A cool property regarding circumcenter

Let $O$ be the circumcenter of $ABC$. Reflect $O$ over $BC$ to obtain $O'$. Through $O'$ construct lines parallel to $AC,AB$ which respectively meet $AB,AC$ at $F,E$. Define $O'F\cap OB=Y, O'E\cap CO=X$. Prove $XY||EF$

I personally think this configuration is very rich and can be exploited to create difficult olympiad geo problems.

#Geometry

Easy Math Editor

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Comments

Notice that $BOCO'$ is a parallelogram. Since $BP || O'C$ and $AB || O'E$ , we see that $\angle FBX = \angle EO'C$ . It is easy to see that $\angle A = \angle BFX = \angle YEC$ . Therefore, $\triangle BFX \sim \triangle O'EC$ . This implies that $\frac{BF}{FX} = \frac{O'E}{EC}$ .

Similarly, we have that $\triangle YCE \sim \triangle BO'F \implies \frac{O'F}{FB} = \frac{CE}{EY}$ .

Multiplying the two ratios completes the proof.

Alan Yan - 5 years, 8 months ago

Yes! The parallel property will hold as long as $BOCO'$ is a parallelogram. There are a couple of typos in your proof, otherwise you got it spot on.

Xuming Liang - 5 years, 8 months ago

PS: The problem can be generalized.

Xuming Liang - 5 years, 8 months ago

Can you post the proof or hint after a few days?

Alan Yan - 5 years, 8 months ago

Yes. :)

Xuming Liang - 5 years, 8 months ago

@Alan Yan Hints: We want to prove some two ratios are equal, perhaps look for some similar triangles from all the parallels.

Generalize this property if you get it.

Xuming Liang - 5 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$