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area of two triangles and line joining their orthocenters

Source: Croatian NMC 2005, 4 th Grade

May 9, 2007

geometryvectorgeometry proposed

Problem Statement

Let

P

and

Q

be points on the sides

BC

and

CD

of a convex quadrilateral

ABCD

, respectively, such that

\angle{BAP}=\angle{ DAQ}

. Prove that the triangles

ABP

and

ADQ

have equal area if and only if the line joining their orthocenters is perpendicular to

AC.