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A convex quadrilateral, bisectors, circumcircles..

Source: Tuymaada 2001, day 2, problem 3.

April 30, 2007

geometrycircumcirclepower of a pointradical axisgeometry proposed

Problem Statement

ABCD

is a convex quadrilateral; half-lines

DA

and

CB

meet at point

Q

; half-lines

BA

and

CD

meet at point

P

. It is known that

\angle AQB=\angle APD

. The bisector of angle

\angle AQB

meets the sides

AB

and

CD

of the quadrilateral at points

X

and

Y

, respectively; the bisector of angle

\angle APD

meets the sides

AD

and

BC

at points

Z

and

T

, respectively. The circumcircles of triangles

ZQT

and

XPY

meet at point

K

inside the quadrilateral. Prove that

K

lies on the diagonal

AC

.
Proposed by S. Berlov

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