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circles & tangents, two lines intersect on circumcircle, irmo geometry 2013

Source: Irmo 2013 p1 q5

September 15, 2018

geometrycircumcirclecircles

Problem Statement

A, B

and

C

are points on the circumference of a circle with centre

O

. Tangents are drawn to the circumcircles of triangles

OAB

and

OAC

P

and

Q

respectively, where

P

and

Q

are diametrically opposite

O

. The two tangents intersect at

K

. The line

CA

meets the circumcircle of

\triangle OAB

A

and

X

. Prove that

X

lies on the line

KO