Is it necessarily true that a triangle is isosceles if the center of the inscribed circle is equidistant from the midpoints of two sides?
Consider a triangle with sides $6, 8, 10$.

## Solution

Consider a right triangle $ABC$ with legs $AB=6, BC=8$ and hypotenuse $AC=10$. Let $O$ be the center of its inscribed circle; $M, N$ be the midpoints of sides $AB$ and $AC$ respectively; $P, Q, F$ be the points of tangency of the inscribed circle with sides $AB, AC$, and $BC$ respectively. Then

$OP=OQ=OF=1/2(AB+BC-AC)=2, PM=BM-BP=1/2 AB-OF=1, QN=AN-AQ=1/2 AC-AP=1$.

Therefore, $OM=ON$.

![](https://cdn.mathpix.com/cropped/2024_05_06_7181633d50815d79481eg-34.jpg?height=603&width=443&top_left_y=307&top_left_x=813)

Answer

Not necessarily.