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

blackpanther Dec 13, 2023

#1**0 **

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.

BuiIderBoi Dec 14, 2023