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Quadrilateral with two opposite angles equal 90, ZIMO - 09

Source: International Zhautykov Olympiad 2009, day 2, problem 5.

January 17, 2009

geometrycyclic quadrilateralsimilar trianglesgeometry proposed

Problem Statement

Given a quadrilateral

ABCD

with \angle B\equal{}\angle D\equal{}90^{\circ}. Point

M

is chosen on segment

AB

so taht AD\equal{}AM. Rays

DM

and

CB

intersect at point

N

. Points

H

and

K

are feet of perpendiculars from points

D

and

C

to lines

AC

and

AN

, respectively. Prove that \angle MHN\equal{}\angle MCK.