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\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v, tetrahedron

Source: Romania IMO TST 1992 p10

February 19, 2020
tetrahedroncoplanar3D geometrygeometryincentertriangle area

Problem Statement

In a tetrahedron VABCVABC, let II be the incenter and A′,B′,C′A',B',C' be arbitrary points on the edges AV,BV,CVAV,BV,CV, and let Sa,Sb,Sc,SvS_a,S_b,S_c,S_v be the areas of triangles VBC,VAC,VAB,ABCVBC,VAC,VAB,ABC, respectively. Show that points A′,B′,C′,IA',B',C',I are coplanar if and only if AA′A′VSa+BB′B′VSb+CC′C′VSc=Sv\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v
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