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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.

Note by Xuming Liang
1 year, 7 months ago

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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 · 1 year, 7 months ago

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@Alan Yan 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 · 1 year, 7 months ago

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@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 · 1 year, 7 months ago

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PS: The problem can be generalized. Xuming Liang · 1 year, 7 months ago

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@Xuming Liang Can you post the proof or hint after a few days? Alan Yan · 1 year, 7 months ago

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@Alan Yan Yes. :) Xuming Liang · 1 year, 7 months ago

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