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AQ/AP wanted, 2 circumcircles each passing through a midpoint

Source: 2007 All-Ukrainian Correspondence MO of magazine ''In the World of Mathematics'', grades 5-11 p11

April 28, 2021

ratiogeometrycircumcircleUkraine Correspondence

Problem Statement

Denote by

B_1

and

C_1

, the midpoints of the sides

AB

and

AC

of the triangle

ABC

. Let the circles circumscribed around the triangles

ABC_1

and

AB_1C

intersect at points

A

and

P

, and let the line

AP

intersect the circle circumscribed around the triangle

ABC

at points

A

and

Q

. Find the ratio

\frac{AQ}{AP}