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Find the ratio of areas of triangles - [UKRMO 2009 Grade 10]

Source:

January 23, 2011

ratiogeometryparallelogramtrigonometrygeometry unsolved

Problem Statement

Let

ABCD

be a parallelogram with

\angle BAC = 45^\circ,

and

AC > BD .

Let

w_1

and

w_2

be two circles with diameters

AC

and

DC,

respectively. The circle

w_1

intersects

AB

E

and the circle

w_2

intersects

AC

O

and

C

, and

AD

F.

Find the ratio of areas of triangles

AOE

and

COF

AO = a,

and

FO = b .