MathDB

7

Part of 2019 Korea Junior Math Olympiad.

Problems(1)

A common ratio related to an acute triangle

Source: Korean junior mathematical olympiad 2019 december

11/17/2019
Let OO be the circumcenter of an acute triangle ABCABC. Let DD be the intersection of the bisector of the angle AA and BCBC. Suppose that ∠ODC=2∠DAO\angle ODC = 2 \angle DAO. The circumcircle of ABDABD meets the line segment OAOA and the line ODOD at E(≠A,O)E (\neq A,O), and F(≠D)F(\neq D), respectively. Let XX be the intersection of the line DEDE and the line segment ACAC. Let YY be the intersection of the bisector of the angle BAFBAF and the segment BEBE. Prove that AY‾BY‾=EX‾EO‾\frac{\overline{AY}}{\overline{BY}}= \frac{\overline{EX}}{\overline{EO}}.
Plane GeometrygeometryKJMOcircumcircleAngle Chasing