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In triangle $ABC,$ let the angle bisectors be $\overline{BY}$ and $\overline{CZ}$.  Given $AB = 12$, $AY = 12$, and $AC = 12$, find $BC$.

 Dec 13, 2023
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We know that triangle ABC is equilateral because each side length is 12.

 

Let I be the incenter of ABC. I is equidistant to segments BY and CZ, which are altitudes of triangle ABC.

 

Using the Power of a Point, we find that AI=AC−12, BI=AB−12, and CI=BC−12. Since AI=BI=CI (triangle AIC is equilateral), it follows that AC=AB=BC. Thus, BC=12​.

 Dec 14, 2023

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