[ Properties and characteristics of an isosceles triangle.] Similarity criteria

Point $M$ lies on the lateral side $A C$ of isosceles triangle $A B C$ with base $B C$, and $B M=B C$. Find $M C$, if $B C=1$ and $A B=2$.
Triangle $M B C$ is isosceles, so $\angle B M C = \angle B C M = \angle C$.

Isosceles triangles $B M C$ and $A B C$ are similar by two angles, therefore, $M C = \frac{B C}{A C} = \frac{1}{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1b324abf723c207e3278g-21.jpg?height=498&width=274&top_left_y=1692&top_left_x=892)

## Answer

$1 / 2$.