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collinear, 2 circles concur on circle - 2015 Cuba 2.6

Source:

September 20, 2024

geometrycollinearconcurrencyconcurrent

Problem Statement

Let

ABC

be a triangle such that

AB > AC

, with a circumcircle

R

. Draw the tangents to

S

at

PR

and

C

and these intersect at

P

. The perpendicular to

AP

through

A

cuts

BC

at

R

. Let

S

be a point on the segment

PR

such that

PS = PC

. (a) Prove that the lines

CS

and

AR

intersect on

\omega

. (b) Let

J

be the midpoint of

J \ne A

and

Q

be the point of intersection of

CS

and

AR

. Circle

Q

and the circumcircle of

\vartriangle AMP

intersect at a point

J

(

J \ne A

), prove that

P

,

J

and

Q

are collinear.

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