The Edge-Tangent Sphere of a Circumscriptible Tetrahedron

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1 Forum Geometricorum Volume 7 (2007) FORUM GEOM ISSN The Edge-Tangent Sphere of a Circumscriptible Tetrahedron Yu-Dong Wu and Zhi-Hua Zhang Abstract A tetrahedron is circumscriptible if there is a sphere tangent to each of its six edges We prove that the radius l of the edge-tangent sphere is at least 3 times the radius of its inscribed sphere This settles affirmatively a problem posed by Z C Lin and H F Zhu We also briefly examine the generalization into higher dimension, and pose an analogous problem for an n dimensional simplex admitting a sphere tangent to each of its edges 1 Introduction Every tetrahedron has a circumscribed sphere passing through its four vertices and an inscribed sphere tangent to each of its four faces A tetrahedron is said to be circumscriptible if there is a sphere tangent to each of its six edges (see [1, ]) We call this the edge-tangent sphere of the tetrahedron Let P denote a tetrahedron P 0 P 1 P 2 P 3 with edge lengths P i P j = a ij fo i<j 3 The following necessary and sufficient condition for a tetrahedron to admit an edge-tangent sphere can be found in [1, 787, 790, 792] See also [4, 6] Theorem 1 The following statement for a tetrahedron P are equivalent (1) P has an edge-tangent sphere (2) a 01 + a 23 = a 02 + a 13 = a 03 + a 12 ; (3) There exist x i > 0, i =0, 1, 2, 3, such that a ij = x i + x j fo i<j 3 For i =0, 1, 2, 3, x i is the length of a tangent from P i to the edge-tangent sphere of P Let l denote the radius of this sphere Theorem 2 [1, 793] The radius of the edge-tangent sphere of a circumscriptible tetrahedron of volume V is given by l = 2x 0x 1 x 2 x 3 (1) 3V Lin and Zhu [4] have given the formula (1) in the form l 2 = 2x 0 x 1 x 2 x 3 (2x 0 x 1 x 2 x 3 ) 2 x i x j (x 2 1 x2 2 x2 3 + x2 2 x2 3 x2 0 + x2 3 x2 0 x2 1 + (2) x2 0 x2 1 x2 2 ) Publication Date: January 22, 2007 Communicating Editor: Paul Yiu The authors would like to thank Prof Han-Fang Zhang for his enthusiastic help

2 20 Y D Wu and Z H Zhang The fact that this latter denominator is (3V ) 2 follows from the formula for the volume of a tetrahedron in terms of its edges: V 2 = (x 0 + x 1 ) 2 (x 0 + x 2 ) 2 (x 0 + x 3 ) (x 0 + x 1 ) 2 0 (x 1 + x 2 ) 2 (x 1 + x 3 ) 2 1 (x 0 + x 2 ) 2 (x 1 + x 2 ) 2 0 (x 2 + x 3 ) 2 1 (x 0 + x 3 ) 2 (x 1 + x 3 ) 2 (x 2 + x 3 ) 2 0 Lin and Zhu op cit obtained several inequalities for the edge-tangent sphere of P They also posed the problem of proving or disproving l 2 3r 2 for a circumscriptible tetrahedron See also [2] The main purpose of this paper is to settle this problem affirmatively Theorem 3 For a circumscriptible tetrahedron with inradius r and edge-tangent sphere of radius l, l 3r 2 Two inequalities Lemma 4 If x i > 0 fo i 3, then ( x1 + x 2 + x 3 + x 2 + x 3 + x 0 + x 3 + x 0 + x 1 + x ) 0 + x 1 + x 2 x 1 x 2 x 3 x 2 x 3 x 0 x 3 x 0 x 1 x 0 x 1 x 2 4(x 0 x 1 x 2 x 3 ) 2 2x 0 x 1 x 2 x 3 x i x j (x 2 1 x2 2 x2 3 + x2 2 x2 3 x2 0 + x2 3 x2 0 x2 1 + x2 0 x2 1 x2 2 ) 6 (3) Proof From x 2 0x 2 1(x 2 x 3 ) 2 + x 2 0x 2 2(x 1 x 3 ) 2 + x 2 0x 2 3(x 1 x 2 ) 2 we have and or +x 2 1 x2 2 (x 0 x 3 ) 2 + x 2 1 x2 3 (x 0 x 2 ) 2 + x 2 2 x2 3 (x 0 x 1 ) 2 0, x 2 1 x2 2 x2 3 + x2 2 x2 3 x2 0 + x2 3 x2 0 x2 1 + x2 0 x2 1 x x 0x 1 x 2 x 3 2x 0 x 1 x 2 x x 0x 1 x 2 x 3 2x 0 x 1 x 2 x 3 x i x j, x i x j (x 2 1 x2 2 x2 3 + x2 2 x2 3 x2 0 + x2 3 x2 0 x2 1 + x2 0 x2 1 x2 2 ) x i x j, 4(x 0 x 1 x 2 x 3 ) 2 x i x j (x 2 1 x2 2 x2 3 + x2 2 x2 3 x2 0 + x2 3 x2 0 x2 1 + x2 0 x2 1 x2 2 ) 4(x 0 x 1 x 2 x 3 ) x = 3x 0x 1 x 2 x 3 0x 1 x 2 x 3 x i x j x i x j (4)

3 The edge-tangent sphere of a circumscriptible tetrahedron 21 On the other hand, it is easy to see that x 1 + x 2 + x 3 x 1 x 2 x 3 + x 2 + x 3 + x 0 x 2 x 3 x 0 + x 3 + x 0 + x 1 x 3 x 0 x 1 + x 0 + x 1 + x 2 x 0 x 1 x 2 = Inequality (3) follows immediately from (4) and (5) 2 x i x j x 0 x 1 x 2 x 3 (5) Corollary 5 For a circumscriptible tetrahedron P with an edge-tangent sphere of radius l, and faces with inradii, r 1, r 2, r 3, ( 1 r r r ) r3 2 l 2 6 Equality holds if and only if P is a regular tetrahedron Proof From the famous Heron formula, the inradius of a triangle ABC of sidelengths a = y + z, b = z + x and c = x + y is given by r 2 = xyz x + y + z Applying this( to the four faces of) P, we see that the first factor on the left hand side of (3) is Now the result follows from (2) r0 2 r1 2 r2 2 r3 2 Proposition 6 Let P be a circumscriptible tetrahedron of volume V If, for i = 0, 1, 2, 3, the opposite face of vertex P i has area i and inradius r i, then ( ) 2 9 V 2 ( 1 2 r r r r3 2 Equality holds if and only if P is a regular tetrahedron ) (6) P 0 Q 3 γ P 1 P 2 α H β Q 2 Q 1 Figure 1 P 3

4 22 Y D Wu and Z H Zhang Proof Let α be the angle between the planes P 0 P 2 P 3 and P 1 P 2 P 3 If the perpendiculars from P 0 to the line P 2 P 3 and to the plane P 1 P 2 P 3 intersect these at Q 1 and H respectively, then P 0 QH = α See Figure 1 Similarly, we have the angles β between the planes P 0 P 3 P 1 and P 1 P 2 P 3, and γ between P 0 P 1 P 2 and P 1 P 2 P 3 Note that P 0 H = P 0 Q 1 sin α = P 0 Q 2 sin β = P 0 Q 3 sin γ Hence, P 0 H P 2 P 3 =2 1 sin α =2 ( cos α)( 1 1 cos α), (7) P 0 H P 3 P 1 =2 2 sin β =2 ( cos β)( 2 2 cos β), (8) P 0 H P 1 P 2 =2 3 sin γ =2 ( cos γ)( 3 3 cos γ) (9) From (7 9), together with P 0 H = 3V 0 and 0 = 1 2 (P 1P 2 + P 2 P 3 + P 3 P 1 ),we have 3V = ( cos α)( 1 1 cos α) + ( cos β)( 2 2 cos β) (10) + ( cos γ)( 3 3 cos γ) Applying Cauchy s inequality and noting that 0 = 1 cos α + 2 cos β + 3 cos γ, we have (3V ) 2 ( cos α cos β cos γ) or ( 1 1 cos α cos β cos γ) (11) =( )( ) =( ) 2 2 0, ( ) 3V 2 ( ) (12) It is easy to see that equality in (12) holds if and only if cos α 1 1 cos α = cos β 2 2 cos β = cos γ 3 3 cos γ Equivalently, cos α =cosβ =cosγ, orα = β = γ Similarly, we have ( ) 3 V 2 ( ) 2 2 1, (13) r 1 ( ) 3 V 2 ( ) 2 2 2, (14) r 2 ( ) 3 V 2 ( ) (15) r 3

5 The edge-tangent sphere of a circumscriptible tetrahedron 23 Summing (12) to (15), we obtain the inequality (6), with equality precisely when all dihedral angles are equal, ie, when P is a regular tetrahedron Remark Inequality (6) is obtained by X Z Yang in [5] 3 Proof of Theorem 3 Since r = 3V , it follows from Proposition 6 and Corollary 5 that l V ( r0 2 r1 2 r2 2 r ) 2 =3r2 This completes the proof of Theorem 3 4 A generalization with an open problem As a generalization of the tetrahedron, we say that an n dimensional simplex is circumscriptible if there is a sphere tangent to each of its edges The following basic properties of a circumscriptible simplex can be found in [3] Theorem 7 Suppose the edge lengths of an n-simplex P = P 0 P 1 P n are P i P j = a ij fo i<j n The n-simplex has an edge-tangent sphere if and only if there exist x i, i =0, 1,,n, satisfying a ij = x i + x j fo i j n In this case, the radius of the edge-tangent sphere is given by where and D 1 = l 2 = D 1 2D 2, (16) 2x 2 0 2x 0 x 1 2x 0 x n 1 2x 0 x 1 2x 2 1 2x 1 x n 1 2x 0 x n 1 2x 1 x n 1 2x 2 n D 2 = D 1 1, We conclude this paper with an open problem: for a circumscriptible n-simplex with a circumscribed sphere of radius R, an inscribed sphere of radius r and an edge-tangent sphere of radius l, prove or disprove that 2n R n 1 l nr

6 24 Y D Wu and Z H Zhang References [1] N Altshiller-Court, Modern Pure Solid Geometry, 2nd edition, Chelsea, 1964 [2] J-C Kuang, Chángyòng Bùděngshì (Applied Inequalities), 3rd ed, Shandong Science and Technology Press, Jinan City, Shandong Province, China, 2004, 252 [3] Z C Lin, The edge-tangent sphere of an n-simplex,(in Chinese) Mathematics in Practice and Theory, 4 (1995) [4] Z C Lin and H F Zhu, Research on the edge-tangent spheres of tetrahedra (in Chinese), Geometric Inequalities in China, pp ; Jiangsu Educational Press, 1996, [5] X Z Yang, On an inequality concerning tetrahedra, (in Chinese), Study in High-School Mathematics, 9 (2005) [6] Z Yang, A necessary and sufficient for a tetrahedron to have an edge-tangent sphere, (in Chinese), Hunan Mathematics Communication, 6 (1985) Yu-Dong Wu: Xinchang High School, Xinchang, Zhejiang Province , P R China address: zjxcwyd@tomcom Zhi-Hua Zhang: Zixing Educational Research Section, Chenzhou, Hunan Province , P R China address: zxzh1234@163com