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Circle through A' and B' touching AB; equal areas

Source: Serbian National Olympiad 2013, Problem 5

April 8, 2013

geometrytrigonometryratiocircumcirclegeometric transformationreflectionparallelogram

Problem Statement

Let

A'

and

B'

be feet of altitudes from

A

and

B

, respectively, in acute-angled triangle

ABC

(

AC\not = BC

). Circle

k

contains points

A'

and

B'

and touches segment

AB

D

. If triangles

ADA'

and

BDB'

have the same area, prove that

\angle A'DB'= \angle ACB.