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Cyclic quadrilateral

Source: Iranian 3rd round Geometry exam P2 - 2014

September 27, 2014

geometrygeometric transformationreflectioncircumcircleangle bisectorgeometry proposed

Problem Statement

\triangle{ABC}

is isosceles

(AB=AC)

. Points

P

and

Q

exist inside the triangle such that

Q

lies inside

\widehat{PAC}

and

\widehat{PAQ} = \frac{\widehat{BAC}}{2}

. We also have

BP=PQ=CQ

.Let

X

and

Y

be the intersection points of

(AP,BQ)

and

(AQ,CP)

respectively. Prove that quadrilateral

PQYX

is cyclic. (20 Points)