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circumcircle passes through the midpoint

Source: Czech-Polish-Slovak Match 2013 day 2 P3

September 29, 2017

geometrycircumcirclereflectionarc midpoint

Problem Statement

Let

{ABC}

be a triangle inscribed in a circle. Point

{P}

is the center of the arc

{BAC}

. The circle with the diameter

{CP}

intersects the angle bisector of angle

{\angle BAC}

at points

{K, L}

{(|AK| <|AL|)}

. Point

{M}

is the reflection of

{L}

with respect to line

{BC}

. Prove that the circumcircle of the triangle

{BKM}

passes through the center of the segment

{BC}

.

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