3. On the sides $AB$ and $BC$ of triangle $ABC$, points $M$ and $N$ are marked respectively such that $\angle CMA = \angle ANC$. Segments $MC$ and $AN$ intersect at point $O$, and $ON = OM$. Find $BC$, if $AM = 5 \, \text{cm}, BM = 3 \, \text{cm}$.
# Solution:

![](https://cdn.mathpix.com/cropped/2024_05_06_de39ee82fbf77ee5c736g-4.jpg?height=248&width=248&top_left_y=2069&top_left_x=961)

Triangles $A M O$ and $C N O$ are congruent ($\angle A M O=\angle C N O, O N=O M, \angle M O A=\angle N O C$, as vertical angles). From the congruence of the triangles, it follows that $A M=N C, \angle M A O=\angle N C O$ and $O A=O C$. Therefore, triangle $A O C$ is isosceles, which means angles $C A O$ and $A C O$ are equal. Hence, triangle $A B C$ is isosceles and $A B=B C=8 \text{ cm}$.

Answer: 8 cm.