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A common ratio related to an acute triangle

Source: Korean junior mathematical olympiad 2019 december

November 17, 2019

Plane GeometrygeometryKJMOcircumcircleAngle Chasing

Problem Statement

Let

O

be the circumcenter of an acute triangle

ABC

. Let

D

be the intersection of the bisector of the angle

A

and

BC

. Suppose that

\angle ODC = 2 \angle DAO

. The circumcircle of

ABD

meets the line segment

OA

and the line

OD

at

E (\neq A,O)

, and

F(\neq D)

, respectively. Let

X

be the intersection of the line

DE

and the line segment

AC

. Let

Y

be the intersection of the bisector of the angle

BAF

and the segment

BE

. Prove that

\frac{\overline{AY}}{\overline{BY}}= \frac{\overline{EX}}{\overline{EO}}

.

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