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sum of areas of triangles ABZ, AYC, XBC equals area of ABC.

Source: Switzerland - Swiss TST 2004 p10

February 18, 2020
triangle areaSumareasgeometryaltitudes

Problem Statement

In an acute-angled triangle ABCABC the altitudes AU,BV,CWAU,BV,CW intersect at HH. Points X,Y,ZX,Y,Z, different from HH, are taken on segments AU,BVAU,BV, and CWCW, respectively. (a) Prove that if X,Y,ZX,Y,Z and HH lie on a circle, then the sum of the areas of triangles ABZ,AYC,XBCABZ, AYC, XBC equals the area of ABCABC. (b) Prove the converse of (a).
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