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tangent from $A$ to the circle meets the line $BC$ at point

Source: JBMO 2005, Problem 2

October 29, 2005

geometrypower of a pointradical axisgeometry proposed

Problem Statement

Let

ABC

be an acute-angled triangle inscribed in a circle

k

. It is given that the tangent from

A

to the circle meets the line

BC

at point

P

. Let

M

be the midpoint of the line segment

AP

and

R

be the second intersection point of the circle

k

with the line

BM

. The line

PR

meets again the circle

k

at point

S

different from

R

. Prove that the lines

AP

and

CS

are parallel.

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