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diagonals of quadrilaterals share intersection points in same ratio

Source: Sharygin 2006 finals 9.6

September 12, 2019
orthocenterdiagonalsratiogeometryconvex quadrilateral

Problem Statement

A convex quadrilateral ABCABC is given. A′,B′,C′,D′A',B',C',D' are the orthocenters of triangles BCD,CDA,DAB,ABCBCD, CDA, DAB, ABC respectively. Prove that in the quadrilaterals ABCPABCP and A′B′C′D′A'B'C'D', the corresponding diagonals share the intersection points in the same ratio.
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