impurity of the corner angles in certain special families of simplices

J. Geom.c© 2014 Springer BaselDOI 10.1007/s00022-014-0219-1 Journal of Geometry

Impurity of the corner angles in certainspecial families of simplices

Mowaffaq Hajja and Mostafa Hayajneh

Abstract. Focusing on the fact that the sum of the angles of any Euclid-ean triangle is constant and equals π for all triangles, Hajja and Mar-tini raised, in [Math Intell 35(3):16–28, 2013, Problem 9], the questionwhether an analogous statement holds for higher dimensional d-simplices.An interesting answer was given by Hajja and Hammoudeh in (Beit Alge-bra Geom (to appear), 2014), where they proved that for the measurearising from what is known as the polar sine, the sum of measures ofthe corner angles of an orthocentric tetrahedron is constant and equalsπ. A crucial ingredient in that treatment is the fact that orthocentric d-simplices are pure, in the sense that the planar subangles of every cornerangle are all acute, all obtuse, or all right. In this article, it is shown thatthis property is not shared by any of the three other special families ofd-simplices that appear in the literature, namely, the families of circum-scriptible, isodynamic, and isogonic (or rather tetra-isogonic) d-simplices,thus answering Problem 3 of (Hajja and Martini in Math Intell 35(3):16–28, 2013). Specifically, it is proved that there are d-simplices in each ofthese families in which one of the corner angles has an acute, an obtuse,and a right planar subangle. The tools used are expected to be useful invarious other contexts. These tools include formulas for the volumes ofd-simplices in these families in terms of the parameters in their standardparameterizations, simple characterizations of the Cayley–Menger deter-minants of such d-simplices, embeddability of a given d-simplex belongingto any of these families in a (d + 1)-simplex in the same family, formulasfor some special determinants, and a nice property of a certain class ofquadratic forms.

Mathematics Subject Classification (2010). Primary 52B11;Secondary 52B12, 52B15, 51M20, 52B10.

Keywords. Cayley–Menger determinant, Cayley–Menger matrix,circumscriptible simplex, corner angle, isodynamic simplex,isogonic simplex, tetra-isogonic simplex, orthocentric simplex,pure polyhedral angle, quadratic form.

M. Hajja and M. Hayajneh J. Geom.

1. Introduction and basic definitions

A characteristic feature of plane Euclidean geometry is the fact that the sum ofmeasures of angles in any triangle is constant, i.e., is the same for all triangles.With this pleasant property in mind, the authors of [19] posed, as Number 9in their list of 14 problems, the question whether this has a higher dimensionalanalogue. Thus they asked if there is a suitable measure for which the sum ofmeasures of corner angles of a d-simplex, d > 2, is constant, where a d-simplexS ⊂ R

n, n ≥ d, is the convex hull of d affinely independent vectors, called thevertices of S. As they felt this to be too good to be true, they suggested thatone may have to restrict oneself to certain dimensions d and to d-simplices ofcertain special type. Even with heavy restrictions of these sorts, one still wouldnot expect a constant angle-sum property for higher dimensional d-simplices. Itwas thus an extremely pleasant surprise when the authors of [20] proved thatif one restricts oneself to d = 3, to orthocentric tetrahedra, i.e., tetrahedrawhose altitudes are concurrent, and to the measure μpolsin arising from what isknown as the polar sine, then the sum of measures of corner angles is constant,and again equals π. It also turns out that the three restrictions are, in somesense, necessary.

The polar sine is one of the three major measures that have appeared often inthe literature, and especially in the recent investigations pertaining to higherdimensional analogues of the pons asinorum theorem, made in [15,16], and [17],and of the open mouth theorem, made in [1], and [18]. If Θ = 〈A;A1, . . . , Ad〉 isthe d-dimensional polyhedral (or simply, the d-polyhedral) angle having vertexA and arms AAi, 1 ≤ i ≤ d, then the polar sine of Θ is defined by

vol(S) =1d!

(d∏

i=1

‖A0 −Ai‖)

polsin Θ, (1)

where S is the d-simplex [A,A1, . . . , Ad], and where vol(S) is its d-volume (ord-dimensional Lebesgue measure). This definition is taken from [10, §1, §7],and it is one of two legitimate generalizations of the ordinary sine. It also hasthe property that 0 ≤ polsin Θ ≤ 1 for all Θ, as can be deduced from theproduct formula (8) in [10]. Motivated by the definition of the measure μ(Θ)of an ordinary (planar) angle Θ given by

μ(Θ) ={

arcsin(sin Θ), if Θ is acute or right,π − arcsin(sin Θ), if Θ is obtuse,

}(2)

one is led to wonder whether d-polyhedral angles, d > 2, can be classified intoacute, right, and obtuse in a natural and useful manner. In this context, it mustcome as pleasant news that corner angles of orthocentric d-simplices are alwayspure in the sense that the planar subangles of any such corner angle Θ are eitherall acute, all obtuse, or all right. One can then refer to a corner angle of anorthocentric d-simplex as acute, obtuse, or right, accordingly. It is also pleasantto know that at most one of the corner angles of an orthocentric d-simplex canbe non-acute, making it possible to classify orthocentric d-simplices as acute,obtuse, or right.

Impurity of the corner angles in certain special families

Using the definition of polsin(Θ) given in (1), it is now clear how to use (2)to define the corresponding measure μpolsin(Θ) of any pure d-polyhedral angleΘ, and hence for any corner angle of an orthocentric d-simplex. The theoremin [20] alluded to earlier states that if Θ1, . . . ,Θd+1 are the corner angles of ad-simplex S, and if ν is a measure on polyhedral angles, then the equation

d+1∑i=1

ν(Θi) is constant (3)

holds if the conditions

(i) S is orthocentric, (ii) ν = μpolsin, and (iii) d = 3 (4)

are satisfied. The questions regarding how necessary the conditions in (4) areto guarantee the conclusion (3) are addressed and treated in [20]. One of thesequestions has to do with Condition (i) of (4) and with whether orthocentricd-simplices can be replaced by any of the well known special families that haveappeared in the literature, namely, the circumscriptible, the isodynamic, andthe isogonic tetrahedra and their d-dimensional analogues. Admitting thatwe see no sensible way for defining μpolsin(Θ) when Θ is not pure, we areforcibly led to ask whether tetrahedra (and more generally d-simplices, d ≥3) that belong to the three afore-mentioned special families are pure, in thesense that their corner angles are. This paper is concerned mainly with thisquestion, and it answers it in the negative by showing that there are, in anyof these special families, d-simplices that have some impure corner angles, i.e.,polyhedral angles whose planar subangles are of mixed types. This emphasizesthe special role that orthocentric d-simplices have among the other families,and shows how indispensable Condition (i) of (4) is for ensuring (3). We addhere that even if we restrict ourselves to the pure tetrahedra that belong toeach of these special families, the result (3) will still not hold.

The paper is organized as follows: In Sect. 2, we describe the Cayley–Mengermatrix of a d-simplex S, and how its determinant is related to the volume ofS. We also describe those matrices that can serve as Cayley–Menger matricesof some d-simplices, and how the Cayley–Menger matrix of a d-simplex Scan be enlarged to a Cayley–Menger matrix of a (d + 1)-simplex obtainedfrom S by adding an extra vertex. In Sect. 3, we find formulas for certainfamilies of determinants, and we use these formulas in Sect. 4 to give a shortproof of the d-dimensional Pythagorean theorem, and in Sect. 5 to deriveusable formulas for the volumes of orthocentric, circumscriptible, isodynamic,and isogonic (or rather tetra-isogonic) d-simplices. In Sect. 6, we establishsome useful properties of certain quadratic forms, and we use these propertiesin Sect. 7 to characterize the Cayley–Menger matrices of d-simplices in thefour special families. In Sect. 8, we show how to embed a given d-simplexof one of these four special types in a (d + 1)-simplex of the same type. InSect. 9, we construct circumscriptible, isodynamic, and isogonic tetrahedrathat are impure. Combined with the results in Sect. 8, this shows how toconstruct impure circumscriptible, isodynamic, and tetra-isogonic d-simplicesfor all d ≥ 3.


2. The Cayley–Menger matrix and determinant

By a (non-degenerate) d-simplex S, we mean the convex hull of d + 1(affinely independent) position vectors (or simply points) A1, . . . , Ad+1 in theEuclidean space R

n for some n ≥ d. The d-simplex S is then denoted byS = [A1, . . . , Ad+1].

Unless explicitly specified, all d-simplices throughout this paper are assumedto be non-degenerate.

If S = [A1, . . . , Ad+1] is a d-simplex, then the Cayley–Menger matrix Γ of S isdefined to be the d+ 2 by d+ 2 symmetric matrix Γ = (xi,j)0≤i,j≤d+1 whosetop row is [0 1 1 · · · 1] and whose other entries are given by

xi,j = ‖Ai −Aj‖2, 1 ≤ i, j ≤ d+ 1. (5)

Thus

Γ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 1 · · · 1 11 0 x1,2 · · · x1,d x1,d+1

1 x2,1 0 · · · x2,d x2,d+1

· · · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · ·1 xd,1 xd,2 · · · 0 xd,d+1

1 xd+1,1 xd+1,2 · · · xd+1,d 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦. (6)

The determinant of Γ is called the Cayley–Menger determinant of S, whichappears in the context of finding a formula for the volume vol(S) of S. Thisformula is given by

2d(d!)2V 2 = (−1)d+1 det(Γ), (7)

and seems to have been discovered for d = 3 (i.e., for the tetrahedron) byTartaglia; see [24, Problem 1.18, p. 29], [27, p. 125], and [22, Formula (1)].

It is convenient to number the rows (and columns) of Γ by the numbers0, 1, 2, . . . , d+ 1. Thus the top row of Γ is referred to as the zero-th (or 0-th)row, and so on. The j-th leading principal minor submatrix of Γ, 0 ≤ j ≤ d+1is the (j + 1) by (j + 1) matrix obtained from Γ by deleting the i-th row andi-th column for all i > j. It is denoted by Γj . Thus Γ = Γd+1. Note that Γj+1

is the Cayley–Menger matrix of the j-simplex [A1, . . . , Aj+1] for 0 ≤ j ≤ d.

The properties that characterize the Cayley–Menger matrix of a d-simplex aregiven in the next theorem. Part (a) is Theorem 9.7.3.4 (p. 239) of [4], and italso appears as Problem 1.22 (p. 30) (with a solution on pp. 215–216) in [24].Part (b) is a trivial consequence of Part (a). We shall see, in Theorem 7.1, thatwhen the matrix Γ given in (6) is of a special type, then Conditions (a) and(b) reduce to the condition (−1)d+1 det(Γ) > 0.

Theorem 2.1. Let xi,j , 1 ≤ i, j ≤ d+ 1, be given real numbers with xi,j = xj,i

for 1 ≤ i, j ≤ d+1. Let Γ be as defined in (6), and let Γi, 0 ≤ i ≤ d+1, denotethe i-th leading principal minor submatrix of Γ. Then Γ is the Cayley–Menger


matrix of some d-simplex if and only if any of the following two conditionsholds:

(a) (−1)i det (Γi) > 0 for 1 ≤ i ≤ d+ 1.(b) Γd is the Cayley–Menger matrix of a (d−1)-simplex, and (−1)d+1 det (Γ)

> 0.

The next theorem shows how the Cayley–Menger matrix of a d-simplex S canbe enlarged to the Cayley–Menger matrix of a (d+1)-simplex obtained from Sby adding an extra vertex. This will be used in Sect. 7 to show how to embeda given d-simplex of one of the four special types in a (d + 1)-simplex of thesame type.

Theorem 2.2. Let xi,j , 1 ≤ i, j ≤ d+ 1, be given real numbers with xi,j = xj,i

for 1 ≤ i, j ≤ d+1. Let Γ be as defined in (6), and let Γi, 0 ≤ i ≤ d+1, denotethe i-th leading principal minor submatrix of Γ. If Γd is the Cayley–Mengermatrix of a (d− 1)-simplex S = [A1, . . . , Ad], then there exists Ad+1 such thatΓ is the Cayley–Menger matrix of the d-simplex [A1, . . . , Ad+1] if and only if(−1)d+1 det(Γ) > 0.

Proof. It follows from Theorem 2.1 that Γ is the Cayley–Menger matrix of ad-simplex T = [B1, . . . , Bd+1] if and only if (−1)d+1 det(Γ) > 0. For such a T ,the d-simplices T ′ = [B1, . . . , Bd] and S have the same Cayley–Menger matrix.It follows that ‖Ai − Aj‖ = ‖Bi − Bj‖ for 1 ≤ i, j ≤ d. Therefore there is,by Theorem 9.7.1 (p. 236) of [4], an (affine) isometry φ : R

d−1 → Rd−1 such

that φ(Bi) = Ai for 1 ≤ i ≤ d. Let ψ : Rd → R

d be an (affine) isometrythat extends φ and let Ad+1 = ψ(Bd+1). Then Ad+1 thus defined satisfies thedesired properties.

Regarding ψ, one defines it by taking e = (0, 0, . . . , 0, 1) ∈ Rd and setting

ψ(e) = e+ φ(0, 0, . . . , 0).

�

3. Some useful determinants

In this section, we derive formulas for the determinants of some matrices thatwill be useful later. These are the matrices J, K, and L introduced in Lem-mas 3.1, 3.3, and 3.4 below. Note that Lemma 3.1 is used in the proof ofLemma 3.3, which in turn is used in the proof of Lemma 3.4.

We start by summarizing the notation and the terminology that will be used. IfM is a square n by n matrix, then its i-th row and i-th column, 1 ≤ i ≤ n, aredenoted by Ri = Ri(M) and Ci = Ci(M), respectively. Sometimes (as in thecase of the Cayley–Menger matrix), we find it more convenient to call the toprow the zero-th row and the leftmost column the zero-th column, and we denotethem by R0 and C0. For 1 ≤ i, j ≤ n, the (i, j)-th minor submatrix of M , i.e.,the matrix obtained from M by deleting the i-th row and the j-th column,is denoted by Mi,j . Its determinant det (Mi,j) is what is usually referred to


as the (i, j)-th minor of M . The operation of interchanging Ri = Ri(M) withRj = Rj(M) will be denoted by Ri ↔ Rj . The operation of moving Ri sothat it lies just after Rj , where j ≥ i, is denoted by Ri �→ Rj,+. Since this isequivalent to performing successively the j − i operations

Ri ↔ Ri+1, Ri+1 ↔ Ri+2, . . . , Rj−1 ↔ Rj ,

it follows that if M ′ is the result of performing the operation Ri �→ Rj,+ on M ,then det(M ′) = (−1)j−i det(M). The operations Ci ↔ Cj and Ci �→ Cj,+ aredefined similarly. These operations can also be performed on determinants.

Lemma 3.1. Let J(n; a, b) be the n by n matrix whose entries mi,j are given by

mi,j = b if i = j, and mi,j = a if i �= j,

and let Ji,j(n; a, b) be the (i, j)-th minor submatrix of J. Then

det (J(n; a, b)) := det

⎡⎢⎢⎢⎢⎣b a · · · · · · aa b · · · · · · a

· · · · · · · · · · · · · · ·· · · · · · · · · · · · · · ·a a · · · · · · b

⎤⎥⎥⎥⎥⎦ (8)

= ((n− 1)a+ b) (b− a)n−1, (9)

det (Ji,j(n; a, b)) =

{((n− 2)a+ b) (b− a)n−2 if i = j,

(−1)j−1−i(b− a)n−2a if i �= j.(10)

Proof. Replacing the first column by the sum of all columns, then replacingevery i-th row Ri, i ≥ 2, by Ri − R1, we obtain a diagonal matrix, and theresult follows.

It remains to prove (10). If i = j, then Ji,j is nothing but J(n − 1; a, b), andthe first line of (10) follows. If i �= j, we may clearly assume that i < j. Inthis case, Ji,j is an n− 1 by n− 1 matrix whose i-th column Ci and (j − 1)-throw Rj−1 have a in each entry. If we perform the operation Ci �→ Cj−1,+, weobtain a matrix D∗ which differs from J(n − 1; a, b) in the (j − 1, j − 1)-thentry only, where D∗ has a in place of b. The (j − 1)-th row R of D∗ (whichconsists of a’s) is the sum of the rows R1, R2 shown in the table

1 2 · · · j − 1 · · · n− 1R a a · · · a · · · aR1 a a · · · b · · · aR2 0 0 · · · a− b · · · 0

.


Letting A1 and A2 be the matrices obtained from D∗ by replacing R with R1

and R2, respectively, we obtain

det (D∗) = det(A1) + det(A2) = det (J(n− 1; a, b))+(a− b) det (J(n− 2; a, b))

= ((n− 2)a+ b) (b− a)n−2 + (a− b)[((n− 3)a+ b) (b− a)n−3

]= (b− a)n−2a,

det(Ji,j) = (−1)j−1−i det(D∗) = (−1)j−1−i(b− a)n−2a.

This proves the second line of (10), and completes the proof of the theorem.

�

Remark 3.2. Formula (9) appears as Problem 192 (p. 35) (with hint on p. 154and answer on p. 187) in [11]. It also appears, in stronger forms, as an exampleon pp. 135–136 of [12] and as Fact 2.12.11 on p. 59 of [5].

Lemma 3.3. Let x = (x1, · · · , xn) and let K(x; a, b) be the n + 1 by n + 1symmetric matrix whose 0-th row is [0, x1, · · · , xn] and whose other entriesmi,j , i �= 0, j �= 0, are given by

mi,j = b if i = j, and mi,j = a if i �= j.

Then

det (K(x, a, b)) := det

⎡⎢⎢⎢⎢⎢⎢⎣

0 x1 x2 · · · · · · xn−1 xn

x1 b a · · · · · · a ax2 a b · · · · · · a a· · · · · · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · · · · · ·xn a a · · · · · · a b

⎤⎥⎥⎥⎥⎥⎥⎦

(11)

= (b− a)n−2[aM2 − ((n− 1)a+ b)N ] , (12)

where

M =n∑

i=1

xi, N =n∑

i=1

x2i . (13)

Proof. Let D = K(x, a, b). Notice that the minor submatrix D0,0 of D isnothing but the matrix J(n; a, b) defined in Theorem 3.1. Let J

(i)(n; a, b)denote the matrix obtained from J(n; a, b) by replacing the i-th columnby the column [x1 x2 . . . xn]t, where t denotes the transpose. Notice thatJk,i(n; a, b) = J

(i)k,i(n; a, b) for 1 ≤ k ≤ n, since J(n; a, b) and J

(i)(n; a, b) differonly in the i-th column. This observation will be freely used.

Expanding det(D) along the 0-th row, and letting Di,j , 0 ≤ i, j ≤ n, be the(i, j)-th minor submatrix of D, we obtain

det(D) =n∑

i=1

(−1)ixi det (D0,i). (14)


On D0,i, we perform the operation C0 �→ Ci−1,+ to obtain

det (D0,i) = (−1)i−1 det(J(i)(n; a, b)

). (15)

We now expand det(J(i)(n; a, b)

)along the i-th column to obtain

det(J(i)(n; a, b)

)=

n∑k=1

xk(−1)k+i det(J(i)k,i(n; a, b)

)

=n∑

k=1

xk(−1)k+i det (Jk,i(n; a, b))

= xi det (Ji,i(n; a, b)) +n∑

k=1, k �=i

xk(−1)k+i det (Jk,i(n; a, b))

= (b− a)n−2

⎡⎣xi((n− 2)a+ b) − a

n∑k=1, k �=i

xk

⎤⎦ .

Therefore,

det(D) =n∑

i=1

(−1)ixi det (D0,i)

=n∑

i=1

(−1)ixi(−1)i−1(b− a)n−2

⎡⎣xi((n− 2)a+ b) − a

n∑k=1, k �=i

xk

⎤⎦

= −(b− a)n−2n∑

i=1

⎡⎣x2

i ((n− 2)a+ b) − a

n∑k=1, k �=i

xixk

⎤⎦

= −(b− a)n−2

⎡⎣((n− 2)a+ b)

n∑i=1

x2i − 2a

∑1≤i<k≤n

xixk

⎤⎦

= −(b− a)n−2[((n− 2)a+ b)N − aM2 + aN ]

= (b− a)n−2[aM2 − ((n− 1)a+ b)N ] ,

as desired. �

Lemma 3.4. For any real numbers s, t and any non-zero real numbers βj , 1 ≤j ≤ d + 1, let Γs,t be the matrix obtained from the matrix Γ defined in (6) bysetting

xij = s(β2i + β2

j ) + tβiβj , i, j ∈ {1, . . . , d+ 1}, i �= j, (16)


i.e.,

Γs,t

=

⎡⎢⎢⎢⎢⎢⎢⎣

0 1 1 · · · 1

1 0 s(β21 + β2

2) + tβ1β2 · · · s(β21 + β2

d+1) + tβ1βd+1

1 s(β22 + β2

1) + tβ2β1 0 · · · s(β22 + β2

d+1) + tβ2βd+1

· · · · · · · · · · · · · · ·1 s(β2

d + β21) + tβdβ1 s(β2

d + β22) + tβdβ2 · · · s(β2

d + β2d+1) + tβdβd+1

1 s(β2d+1 + β2

1) + tβd+1β1 s(β2d+1 + β2

2) + tβd+1β2 · · · 0

⎤⎥⎥⎥⎥⎥⎥⎦

(17)

Let

M =d+1∑i=1

1βi, N =

d+1∑i=1

1β2

i

, P =d+1∏i=1

βi. (18)

Then

det(Γs,t) = (−1)d+1P2(2s+ t)d−1[tM2 − (dt− 2s)N ]. (19)

Proof. Denote the i-th row and the i-th column of M, 0 ≤ i ≤ d+ 1, as usual,by Ri and Ci, respectively. Replacing Ci, 1 ≤ i ≤ d + 1, by Ci − sβ2

i C0, andthen replacing Ri, 1 ≤ i ≤ d+ 1, by Ri − sβ2

iR0, we obtain

det(Γs,t) = det

⎡⎢⎢⎢⎢⎢⎢⎣

0 1 1 · · · 1 11 −2sβ2

1 tβ1β2 · · · tβ1βd tβ1βd+1

1 tβ2β1 −2sβ22 · · · tβ2βd tβ2βd+1

· · · · · · · · · · · · · · · · · ·1 tβdβ1 tβdβ2 · · · −2sβ2

d tβdβd+1

1 tβd+1β1 tβd+1β2 · · · tβd+1βd −2sβ2d+1

⎤⎥⎥⎥⎥⎥⎥⎦.

Letting γj = 1/βj , 1 ≤ j ≤ d+ 1, we see that

det(Γs,t) = P2 det(D), (20)

where

D =

⎡⎢⎢⎢⎢⎢⎢⎣

0 γ1 γ2 · · · γd γd+1

γ1 −2s t · · · t tγ2 t −2s · · · t t· · · · · · · · · · · · · · · · · ·γd t t · · · −2s tγd+1 t t · · · t −2s

⎤⎥⎥⎥⎥⎥⎥⎦. (21)

Thus D is nothing but the matrix K(x, t,−2s) defined in Lemma 3.3, wherex = (γ1, . . . , γd+1). By Lemma 3.3,

det(D) = (−2s− t)d−1[tM2 − (dt− 2s)N ].

Using this and (20), we obtain the desired result. �

Remark 3.5. Although the quantities M and N , defined in (18), are not definedif any βi is 0, the quantities PM and P2N are defined for such values, sinceno βi appears in their denominators. Hence the same holds for the right handside of (19).


4. Volume of a rectangular d-simplex and a proof of the higherdimensional Pythagorean theorem

A d-simplex S = [A1, . . . , Ad+1] is said to be rectangular at Ad+1 if∠AiAd+1Aj = 90◦ for all i, j ∈ {1, . . . , d} with i �= j. Rectangular d-simplicesare clearly the natural analogues of right-angled triangles, and they are knownto admit the d-dimensional Pythagorean relation

f2d+1 = f2

1 + · · · + f2d ,

where fj is the (d − 1)-volume of the j-th facet. A proof can be found in[9,23,26], and [3]. According to [6], it is very likely that R. Descartes knewthe three dimensional version, which also appears in [13, pp. 911–912], whereit is attributed to J.-P. Gua de Malves who published it in his memoirs of1873. This version appears as an exercise in [25], and in [2, Section 11.1, pp.192–193], where it is called de Gua’s theorem in. In this section, we shall usethe results of the previous section to give a short proof of the d-dimensionalversion.

Let S = [A1, . . . , Ad+1] be a rectangular d-simplex, with its right angle at Ad+1.Let ‖Ad+1 −Ai‖ = βi for all i ∈ {1, . . . , d}. Setting βd+1 = 0, we see that theedge lengths of S are given by ‖Ai −Aj‖2 = β2

i +β2j for all i, j ∈ {1, . . . , d+1}

with i �= j. Thus the Cayley–Menger matrix of S is the matrix denoted, inLemma 3.4, by Γ1,0 and is hence equal to

det(Γ1,0) = (−1)d+1(2d)

(d∏

i=1

βi

)2

. (22)

Note that taking βd+1 = 0 is legitimized bt Remark 3.5. Using (7), we see thatthe d-volume ‖S‖ of S is given by

‖S‖ =(

1d!

)( d∏i=1

βi

). (23)

Now let Si, 1 ≤ i ≤ d + 1, be the i-th facet of S, and let ‖Si‖ be its(d− 1)-volume. By the ordinary Pythagoras’ theorem, the edge lengths of thehypotenuse facet Sd+1 are given by ‖Ai−Aj‖2 = β2

i +β2j for all i, j ∈ {1, . . . , d}

with i �= j. Thus the Cayley–Menger matrix of Sd+1 is the (d+ 1) by (d+ 1)matrix denoted, in Lemma 3.4, by Γ1,0, with the understanding that d + 1 isto be replaced by d, and β1, . . . , βd+1 by β1, . . . , βd. Its determinant is thusgiven by

det(Γ1,0) = (−1)d

(d∏

i=1

βi

)2

2d−1

(d∑

i=1

1β2

i

). (24)

Using (7), we see that

‖Sd+1‖2 =(

1(d− 1)!

)2(

d∏i=1

βi

)2( d∑i=1

1β2

i

). (25)


On the other hand,

d∑i=1

‖Si‖2 =d∑

i=1

(1

(d− 1)!

)2⎛⎝ d∏

j=1, j �=i

βj

⎞⎠

2

(26)

=(

1(d− 1)!

)2⎛⎝ d∏

j=1

βj

⎞⎠

2(d∑

i=1

1β2

i

). (27)

Therefore

‖Sd+1‖2 =d∑

i=1

‖Si‖2. (28)

We have thus proved the d-dimensional Pythagorean theorem.

5. Special simplices and their volumes

A tetrahedron is called orthocentric if its altitudes are concurrent; it is calledcircumscriptible if it admits a sphere that touches its edges; it is called isody-namic if the cevians joining the vertices to the incenters of the respective facesare concurrent; and it is called isogonic if the cevians joining the vertices to thepoints where the insphere touches the respective faces are concurrent. Thesefour types of tetrahedra are studied in detail in [7, Chapter IX.B, pp. 294–333],and they have been appearing every now and then in articles and in problemproposals since then. The definitions of the first three families extend natu-rally and unequivocally to higher dimensions, where one can safely declare ad-simplex S to be orthocentric (respectively, circumscriptible, isodynamic) ifevery four vertices of S form an orthocentric (respectively, circumscriptible,isodynamic) tetrahedron. The same cannot be said of isogonic d-simplices, andthe different possibilities for defining them for d ≥ 4 do not seem to be equiva-lent. These issues are discussed in detail in [21], where the authors have givenvarious old and new characterizations of the different types of d-simplices, andwhere they have chosen to call a d-simplex S tetrahedron isogonic (or simplytetra-isogonic) if every four vertices of S form an isogonic tetrahedron.

In this section, we derive formulas for the volumes of these four types of d-simplices. We start with Theorem 5.1 giving an algebraic characterization,taken from [21], of each of these types of d-simplices. Then we derive, in The-orem 5.2, formulas for their Cayley–Menger determinants. For their volumes,one uses Formula (7). The characterizations in Theorem 5.1 can be thought ofas definitions.

Theorem 5.1. Let S = [A1, . . . , Ad+1] be a d-simplex. Then

S is orthocentric ⇐⇒ ∃ βi ∈ R with ‖Ai −Aj‖2 = βi + βj , (29)S is circumscriptible ⇐⇒ ∃ βi > 0 with ‖Ai −Aj‖ = βi + βj , (30)

S is isodynamic ⇐⇒ ∃ βi > 0 with ‖Ai −Aj‖2 = βiβj , (31)


S is tetra-isogonic ⇐⇒ ∃ βi > 0 with ‖Ai −Aj‖2 = β2i + βiβj + β2

j ,

(32)

where the indices in the right hand sides run over all i, j ∈ {1, . . . , d+ 1} withi �= j. Also, the parameters βi, 1 ≤ i ≤ d+1, appearing in the right hand sidesare unique.

Theorem 5.2. Let S(o), S(c), S(y), S(g) be the orthocentric, circumscriptible,isodynamic, and tetra-isogonic d-simplices S = [A1, . . . , Ad+1] defined by (29),(30), (31), and (32), and let their Cayley–Menger matrices be Γ(o), Γ(c), Γ(y),Γ(g), respectively. Let M, N , and P be defined as in (18). Then

det(Γ(o)) = (−1)d+1 2d P M, (33)

det(Γ(c)) = (−1)d+1 22d−1 P2 (M2 − (d− 1)N ), (34)

det(Γ(y)) = (−1)d+1 P2 (M2 − dN ), (35)

det(Γ(g)) = (−1)d+1 3d−1P2 (M2 − (d− 2)N ). (36)

Proof. The last three formulas follow immediately from Lemma 3.4 by taking(s, t) to be (1, 2), (0, 1), and (1, 1), respectively. Note that (30) can be writtenas ‖Ai−Aj‖2 = (β2

i +β2j )+2βiβj . The first formula also follows from Lemma 3.4

by taking (s, t) to be (1, 0) and by replacing β2i by βi. Note that if we substitute

βi for β2i in N and in P2, we obtain M and P, respectively. �

6. Properties of certain quadratic forms

In this section, we exhibit a nice property of a certain type of quadratic form.This will be used later in the proof of Theorems 7.1 and 8.1.

Lemma 6.1. Let s ∈ R. For k ∈ N, let the quadratic form Qk = Q(s)k : R

k → R

be defined by

Qk(z1, . . . , zk) = (z1 + · · · + zk)2 − (k − s)(z21 + · · · + z2

k). (37)

(a) If Qk+1(t1, . . . , tk+1) > 0 for some t1, . . . , tk+1 > 0, then Qj(t1, . . . , tj) >0 for 1 ≤ j ≤ k + 1.

(b) If Qk(t1, . . . , tk) > 0 for some t1, . . . , tk > 0, then there exists an openinterval W in (0,∞) such that Qk+1(t1, . . . , tk+1) > 0 for all tk+1 ∈ W .

Proof. To prove (a), we let t1, . . . , tk+1 > 0 be such that Qk+1(t1, . . . , tk+1) >0, and we prove that Qk(t1, . . . , tk) > 0. Set

m = t1 + · · · + tk, n = t21 + · · · + t2k. (38)

Then

Qk+1(t1, . . . , tk+1) = (t1 + · · · + tk+1)2 − (k + 1 − s)(t21 + · · · + t2k+1)

= (tk+1 +m)2 − (k + 1 − s)(t2k+1 + n)

= (−k + s)t2k+1 + 2mtk+1 +m2 − (k + 1 − s)n. (39)


Notice that if k− s ≤ 0, then Qk(t1, . . . , tk) = m2 − (k− s)n > 0, and there isnothing to prove. Thus we may assume that

k − s > 0. (40)

Then the parabola (in U) defined by

f(U) = (−k + s)U2 + 2mU +m2 − (k + 1 − s)n (41)

opens down. Also, its value at U = tk+1 is Qk+1 (t1, . . . , tk+1) by (39), and ishence positive. Therefore its discriminant Δ is positive. But

Δ = 4[m2 − (−k + s)(m2 − (k + 1 − s)n)

]= 4[(1 + k − s)m2 + (−k + s)(k + 1 − s)n

]= 4(1 + k − s)

[m2 + (−k + s)n

]= 4(1 + k − s)

[(t1 + · · · + tk)2 − (k − s)(t21 + · · · + t2k)

]= 4(1 + k − s)Qk(t1, . . . , tk). (42)

Since Δ > 0 and 1 + k − s > 0 (by (40)), it follows that Qk(t1, . . . , tk) > 0, asdesired. This proves (a).

To prove (b), we let t1, . . . , tk > 0 be such that Qk(t1, . . . , tk) > 0, and weprove that Qk+1(t1, . . . , tk+1) > 0 for all tk+1 in some open interval W in(0,∞).

Let m, n be as in (38), and consider the parabola (in U) defined in (41). By(39),

Qk+1(t1, . . . , tk+1) = f(tk+1).

If s− k > 0, then the parabola in (41) opens up, and f(tk+1) > 0 for all tk+1

large enough.

If s− k < 0, then the parabola in (41) opens down. Its discriminant, given by(42), is positive, since 1+k−s > 0. Also the sum of its zeros is s = −2m/(−k+s), and hence f(s/2) = f(−m/(−k + s)) > 0. Therefore f(tk+1) > 0 for alltk+1 in some neighborhood of −m/(−k + s).

If s−k = 0, then f(U) = 2mU+m2 −n. This is positive for all U > 0, because

m2 − n = (t1 + · · · + tk)2 − (t21 + · · · + t2k) =∑

1≤i<j≤k

2titj > 0.

Thus f(tk+1) > 0 for all tk+1 > 0.

Thus in all cases, there is an open interval W in (0,∞) such that f(tk+1) > 0for all tk+1 ∈ W .

This completes the proof of (b). �


7. Restrictions on the values of βi in Theorem 5.1

The next theorem describes the restrictions on the values of β1, . . . , βd+1 thatappear in (29), (30), (31), and (32). It is a stronger version of Theorem 2.1 thatholds for the special matrices appearing as Cayley–Menger matrices of ortho-centric, circumscriptible, isodynamic, and tetra-isogonic d-simplices. The partpertaining to circumscriptible d-simplices provides another proof of Theorem4.10 of [14].

Theorem 7.1. Let β1, . . . , βd+1 be given real numbers, and let Γ(o), Γ(c), Γ(y),and Γ(g) be the matrices obtained from the matrix Γ defined in (6) by settingxij equal to

βi + βj , (βi + βj)2, βiβj , β2i + βiβj + β2

j ,

respectively. Thus Γ(o), Γ(c), Γ(y), and Γ(g) are given by

Γ(o) =

⎡⎢⎢⎢⎢⎢⎢⎣

0 1 1 · · · 1 11 0 β1 + β2 · · · β1 + βd β1 + βd+1

1 β2 + β1 0 · · · β2 + βd β2 + βd+1

· · · · · · · · · · · · · · · · · ·1 βd + β1 βd + β2 · · · 0 βd + βd+1

1 βd+1 + β1 βd+1 + β2 · · · βd+1 + βd 0

⎤⎥⎥⎥⎥⎥⎥⎦, (43)

Γ(c)

=

⎡⎢⎢⎢⎢⎢⎢⎣

0 1 1 · · · 1 11 0 (β1 + β2)2 · · · (β1 + βd)2 (β1 + βd+1)2

1 (β2 + β1)2 0 · · · (β2 + βd)2 (β2 + βd+1)2

· · · · · · · · · · · · · · · · · ·1 (βd + β1)2 (βd + β2)2 · · · 0 (βd + βd+1)2

1 (βd+1 + β1)2 (βd+1 + β2)2 · · · (βd+1 + βd)2 0

⎤⎥⎥⎥⎥⎥⎥⎦,

(44)

Γ(y) =

⎡⎢⎢⎢⎢⎢⎢⎣

0 1 1 · · · 1 11 0 β1β2 · · · β1βd β1βd+1

1 β2β1 0 · · · β2βd β2βd+1

· · · · · · · · · · · · · · · · · ·1 βdβ1 βdβ2 · · · 0 βdβd+1

1 βd+1β1 βd+1β2 · · · βd+1βd 0

⎤⎥⎥⎥⎥⎥⎥⎦, (45)

Γ(g)

=

⎡⎢⎢⎢⎢⎢⎢⎣

0 1 1 · · · 11 0 β2

1 + β1β2 + β22 · · · β2

1 + β1βd+1 + β2d+1

1 β22 + β2β1 + β2

1 0 · · · β22 + β2βd+1 + β2

d+1

· · · · · · · · · · · · · · ·1 β2

d + βdβ1 + β21 β2

d + βdβ2 + β22 · · · β2

d + βdβd+1 + β2d+1

1 β2d+1 + βd+1β1 + β2

1 β2d+1 + βd+1β2 + β2

2 · · · 0

⎤⎥⎥⎥⎥⎥⎥⎦

.

(46)

Let M, N , and P be defined as in (18). Then


(a)

Γ(o) is the Cayley–Menger matrix of an orthocentric d-simplex

⇐⇒ (i)βi + βj > 0 for 1 ≤ i < j ≤ d+ 1, and (ii) (−1)d+1 det(Γ(o)) > 0⇐⇒ (i)βi + βj > 0 for 1 ≤ i < j ≤ d+ 1, and (ii) PM > 0.

(b) Suppose that βi > 0 for 1 ≤ i ≤ d+ 1. Then

Γ(c) is the Cayley–Menger matrix of a circumscriptible d-simplex

⇐⇒ (−1)d+1 det(Γ(c)) > 0 ⇐⇒ M2 − (d− 1)N > 0,

Γ(y) is the Cayley–Menger matrix of an isodynamic d-simplex

⇐⇒ (−1)d+1 det(Γ(y)) > 0 ⇐⇒ M2 − dN > 0.

Γ(g) is the Cayley–Menger matrix of a tetra-isogonic d-simplex

⇐⇒ (−1)d+1 det(Γ(g)) > 0 ⇐⇒ M2 − (d− 2)N > 0.

Proof. We start by giving a combined proof of the three parts of (b). We let Mstand for Γ(c), Γ(y), or Γ(g), as the case may be. We also let Mi, 1 ≤ i ≤ d+1,be the i-th leading principal minor submatrix of M . Then

M is the Cayley–Menger matrix of a d-simplex⇐⇒ (−1)i det(Mi) > 0 for 1 ≤ i ≤ d+ 1 (by Theorem 2.1)⇐⇒ M2

i − (i− s)Ni > 0 for 1 ≤ i ≤ d+ 1,

where s = 2 if M = Γ(c), s = 1 if M = Γ(y), s = 3 if M = Γ(g),

by (34), (35), (36)⇐⇒ M2

d+1 − (d+ 1 − s)Nd+1 > 0 (by Lemma 6.1 (a))

⇐⇒ (−1)d+1 det(M) > 0,

as desired.

It remains to prove (a). We again let Mi, 1 ≤ i ≤ d + 1, be the i-th leadingprincipal minor submatrix of M , and we put γi = 1/βi for 1 ≤ i ≤ d+ 1. The“only if” part is trivial by the definition of the Cayley–Menger matrix and by(2). For the “if”part, we take two cases.

Case 1. If βi > 0 for all i, then the assumptions (i) and (ii) are redundant,since

det(Mj) =

(j∏

i=1

βi

)(j∑

i=1

γi

)

is obviously positive for all j, 1 ≤ j ≤ d+ 1. So we are done by Theorem 2.1.

Case 2. If βk ≤ 0 for some k, 1 ≤ k ≤ d+ 1, then βi > 0 for all i �= k, becauseβk + βi > 0. In this case, we have

d+1∏i=1

βi < 0,d+1∑i=1

γi < 0, by (ii).


If j < k, thenj∏

i=1

βi > 0,j∑

i=1

γi > 0,

and hence det(Mj) > 0. If j ≥ k, thenj∏

i=1

βi < 0,j∑

i=1

γi <

d+1∑i=1

γi < 0,

and therefore det(Mj) > 0 again. Thus det(Mj) > 0 for all j, 1 ≤ j ≤ d, andagain we are done by Theorem 2.1. as desired. �

8. Embedding a d-simplex of a certain type in a(d + 1)-simplex of the same type

In this section, we prove that every orthocentric (respectively, circumscriptible,isodynamic, tetra-isogonic) d-simplex can be embedded in a (d+1)-simplex ofthe same type.

Theorem 8.1. Let S = [A1, . . . , Ad] be an orthocentric (respectively, circum-scriptible, isodynamic, tetra-isogonic) (d − 1)-simplex in the Euclidean spaceR

d−1, where d ≥ 3. Then there exist points Ad+1 in Rd such that the d-simplex

[A1, . . . , Ad, Ad+1] is orthocentric (respectively, circumscriptible, isodynamic,tetra-isogonic).

Proof. We give a unified proof for the circumscriptible, isodynamic, and tetra-isogonic cases.

By Theorem 5.1, there exist β1, . . . , βd > 0 such that

‖Ai −Aj‖2 =

⎧⎨⎩

(βi + βj)2 if S is circumscriptibleβiβj if S is isodynamicβ2

i + βiβj + β2j if S is tetra-isogonic

(47)

for 1 ≤ i < j ≤ d. By Theorem 2.2, it is sufficient to find βd+1 > 0 for whichthe respective matrix Γ defined in (44), (45), or (46) has the property that(−1)d+1 det(Γ) > 0. Let

s =

⎧⎨⎩

2 if S is circumscriptible1 if S is isodynamic3 if S is tetra-isogonic

,

and let γi = 1/βi for 1 ≤ i ≤ d, as usual. Using (34), (35), (36), and definingthe quadratic form Qk, k ∈ N, as in (37) by

Qk(z1, . . . , zk) = (z1 + · · · + zk)2 − (k − s)(z21 + · · · + z2

k

),

we see that

(−1)d+1 det(Γ) > 0 ⇐⇒ (γ1 + · · · + γd+1)2 − (d+ 1 − s)

(γ21 + · · · + γ2

d+1

)⇐⇒ Qd+1(γ1, . . . , γd+1) > 0.


But we know that Qd(γ1, . . . , γd) > 0, since β1, . . . , βd define a (d−1)-simplex.Therefore it follows from Part (b) of Lemma 6.1 that there is an open intervalW in (0,∞) for which Qd+1(γ1, . . . , γd+1) > 0 for all γd+1 in W . This openinterval corresponds to an open interval containing βd+1. This proves the caseswhen S is circumscriptible, isodynamic, or tetra-isogonic.

It remains to deal with the case when S is orthocentric. In this case, thereexist, by Theorem 5.1, β1, . . . , βd ∈ R such that

‖Ai −Aj‖2 = βi + βj for 1 ≤ i < j ≤ d. (48)

By Theorem 7.1, we know that

βi + βj > 0 for 1 ≤ i ≤ d, (49)

(β1 · · ·βd)(γ1 + · · · + Γ(y)

)> 0, (50)

where γi = 1/βi for 1 ≤ i ≤ d. We are to find βd+1 ∈ R such that

βi + βj > 0 for 1 ≤ i ≤ d+ 1. (51)

(β1 · · ·βdβd+1)(γ1 + · · · + Γ(y) + γd+1

)> 0, (52)

where γd+1 = 1/βd+1.

If β1, . . . , βd > 0, then we choose βd+1 to be any positive number. If βk < 0for some k, 1 ≤ k ≤ d, then ρ := −(γ1 + · · · + Γ(y)) > 0, by (50) and (49), andwe can choose γd+1 to be any number in (0, ρ). For such a choice,

γd+1 < −(γ1 + · · · + Γ(y)) < −γk, (53)

and hence γd+1 + γk < 0. Therefore βd+1 + βk > 0. The remaining sumsβi + βj , 1 ≤ i < j ≤ d+ 1, are positive because of (49) and because βj > 0 forj �= k. As for (52), it follows immediately from (53). �

9. Constructing impure special simplices

In this section, we show how to construct circumscriptible, isodynamic, andtetra-isogonic d-simplices, d ≥ 3, each having a corner angle with an acute,right, and obtuse planar subangle. This shows that d-simplices of these threetypes are not necessarily pure, and contrasts heavily with the case of ortho-centric d-simplices.

In view of Theorem 8.1, it is sufficient to restrict ourselves to d = 3, i.e., totetrahedra. Thus we write “isogonic” for “tetra-isogonic”.

9.1. Constructing impure circumscriptible tetrahedra

We shall construct a circumscriptible tetrahedron XY ZW in which one of thecorner angles consists of one acute, one right, and one obtuse planar angle.


Figure 1 Illustrating the planar subangles ξ, η, and ζ of thecorner angle W

Let XY ZW be a general circumscriptible tetrahedron, and let x, y, z, w be thepositive numbers, guaranteed by Theorem 5.1, for which

‖X − Y ‖ = x+ y, ‖Y − Z‖ = y + z, ‖Z −X‖ = z + x,

‖W −X‖ = w + x, ‖W − Y ‖ = w + y, ‖W − Z‖ = w + z. (54)

By Theorem 7.1, the only condition that (the positive numbers) x, y, z, w haveto satisfy is

Q :=(

1x

+1y

+1z

+1w

)2

− 2(

1x2

+1y2

+1z2

+1w2

)> 0. (55)

Let ξ = ∠YWZ, η = ∠ZWX, and ζ = ∠XWY , as shown in Fig. 1. Withoutloss of generality, we assume that w = 1. We also set

1 +1x

= L, 1 +1y

= M, 1 +1z

= N. (56)

Using the law of cosines, we see that

cos ξ > 0 ⇐⇒ 1 + y + z − yz > 0 ⇐⇒(

1 +1y

)(1 +

1z

)> 2.

Thus we have

cos ξ > 0 ⇐⇒ MN > 2, cos ξ = 0 ⇐⇒ MN = 2, cos ξ < 0 ⇐⇒ MN < 2.

Similar statements hold for cos η and cos ζ.

To arrange for ξ to be right, η acute, and ζ obtuse, we look for ρ ∈ (0, 1) forwhich MN = 2, NL = 2 + ρ, and LM = 2 − ρ. For this, we have to take

L =

√(2 + ρ)(2 − ρ)

2, M =

√2(2 − ρ)2 + ρ

, N =

√2(2 + ρ)2 − ρ

. (57)


The requirement x, y, z > 0 is equivalent to L,M,N > 1, which in turn isequivalent to ρ < 2/3. The requirement (55) simplifies, using (56) and (57)and routine calculations, to the condition ρ4 + 16ρ2 − 16 < 0, or equivalentlyto the condition ρ2 < 4

√5 − 8 ≈ 0.944. This is obviously satisfied if ρ < 2/3.

Thus taking any ρ ∈ (0, 23

), and letting x, y, z be defined by (56) and (57), we

see that the corner angle at W of the circumscriptible tetrahedron XY ZWdefined by (54) consists of an acute angle, a right angle, and an obtuse angle,as desired.

9.2. Constructing impure isodynamic tetrahedra

Let us now assume that XY ZW is isodynamic. Then there exist positivenumbers x, y, z, w such that

‖X − Y ‖2 = xy, ‖Y − Z‖2 = yz, ‖Z −X‖2 = zx,

‖W −X‖2 = wx, ‖W − Y ‖2 = wy, ‖W − Z‖2 = wz. (58)

By Theorem 7.1, the only condition that the positive numbers x, y, z, w haveto satisfy is

Q :=(

1x

+1y

+1z

+1w

)2

− 3(

1x2

+1y2

+1z2

+1w2

)> 0. (59)

We again define ξ, η, ζ as shown in Fig. 1, and we again assume w = 1. Thistime we set

1x

= L,1y

= M,1z

= N. (60)

Then

cos ξ > 0 ⇐⇒ y + z − zy > 0 ⇐⇒ 1y

+1z> 1.

Thus

cos ξ>0 ⇐⇒ M +N>1, cos ξ=0 ⇐⇒ M +N = 1, cos ξ<0 ⇐⇒ M +N<1.


To arrange for ξ, η, ζ to be right, acute, obtuse, respectively, we need to findρ ∈ (0, 1) for which M +N = 1, N +L = 1 + ρ, and L+M = 1 − ρ. Thus wehave to take

L =12, M =

12

− ρ, N =12

+ ρ, (61)

showing that we need to take ρ such that ρ < 1/2. By (60), the condition (59)simplifies into ρ2 < 1/6. Once this is satisfied, the previous condition ρ < 1/2is also satisfied.

Thus taking any ρ ∈(0,√

1/6), and letting x, y, z be defined by (60) and

(61), we see that the corner angle at W of the isodynamic tetrahedron XY ZWdefined by (58) consists of an acute angle, a right angle, and an obtuse angle.


9.3. Constructing impure isogonic tetrahedra

Let us now assume that XY ZW is isogonic. By Theorem 5.1, there existpositive numbers x, y, z, w such that

‖X − Y ‖2 = x2 + xy + y2, ‖Y − Z‖2 = y2 + yz + z2,

‖Z −X‖2 = z2 + zx+ x2, ‖W −X‖2 = w2 + wx+ x2,

‖W − Y ‖2 = w2 + wy + y2, ‖W − Z‖2 = w2 + wz + z2. (62)

By Theorem 7.1, the only condition that (the positive numbers) x, y, z, w haveto satisfy is

Q :=(

1x

+1y

+1z

+1w

)2

−(

1x2

+1y2

+1z2

+1w2

)> 0. (63)

This condition is vacuous, since the right hand side of (63) is a sum of positiveterms.

We again define ξ, η, ζ as shown in Fig. 1, and we again assume w = 1. Thistime we set

x− 1 = L, y − 1 = M, z − 1 = N. (64)

Using the law of cosines, we see that

cos ξ > 0 ⇐⇒ 2 + y + z − yz > 0 ⇐⇒ (y − 1)(z − 1) < 3.

Thus

cos ξ > 0 ⇐⇒ MN < 3, cos ξ = 0 ⇐⇒ MN = 3, cos ξ < 0 ⇐⇒ MN > 3.


To arrange for ξ, η, ζ to be right, acute, obtuse, respectively, we need to findρ > 0 for which MN = 3, NL = 3−ρ, and LM = 3+ρ. Thus we have to take

L =

√(3 + ρ)(3 − ρ)

3, M =

√3(3 + ρ)3 − ρ

, N =

√3(3 − ρ)3 + ρ

, (65)

showing that we need to take ρ such that ρ < 3.

Thus taking any ρ ∈ (0, 3), and letting x, y, z be defined by (64) and (65), wesee that the corner angle at W of the isogonic tetrahedron XY ZW defined by(62) consists of an acute angle, a right angle, and an obtuse angle.

Acknowledgments

The financial support of Yarmouk University is acknowledged. M. Hayajnehwould like to thank his friends H. Dang and B. Nguyen for the discussions.


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Mowaffaq HajjaDepartment of MathematicsYarmouk UniversityIrbid, Jordane-mail: [email protected];

[email protected]

Mostafa HayajnehDepartment of MathematicsLouisiana State UniversityBaton RougeLA 70803USAe-mail: [email protected]

Received: December 5, 2013.

Revised: March 14, 2014.

http://students.imsa.edu/~tliu/Math/planegeo.pdf

http://students.imsa.edu/~tliu/Math/planegeo.pdf

impurity of the corner angles in certain special families of simplices

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