000RM.dvi

1108 Tetrahedra

42.2 The volume of an isosceles tetrahedron

LetLbe the pedal of the vertexDon the faceABC.^1 Consider the
plane throughDperpendicular to the facesABCandDBC. This is the
plane containingD,L, and the common pedalXof these points on the
lineBC. Upon unfolding the faceDABinto triangleABC′, triangle
DZLbecomes the segmentC′LintersectingABatZ. SinceC′Zis
perpendicular toAB,C′Lis perpendicular toA′B′. The same reasoning
applied to the other two facesDBCandDCAshows thatA′L,B′L,
C′Lare perpendicular toB′C′,C′A′,A′B′respectively. It follows that
Lis the orthocenter of triangleA′B′C′.

A

B C

D

C′

L

Z

A

B C

C′ D

Z L

B′

A′

HO

G

Proposition 42.1.The pointLis the reflection ofHinO.

The pointLis called the de Longchamps point of triangleABC.

Proposition 42.2.

OL^2 =OH^2 =R^2 (1−8cosαcosβcosγ).

Theorem 42.3.The volume of the isosceles tetrahedron on triangleABC
is given by

Viso=

√

1

72

(b^2 +c^2 −a^2 )(c^2 +a^2 −b^2 )(a^2 +b^2 −c^2 ).

(^1) We use the wordpedalforperpendicular footororthogonal projection.