4. The altitudes $C D$ and $A E$ of triangle $A B C$ intersect at point $H$ and it is given that $d(A, B)=d(C, H)$. Determine $\measuredangle A C B$.
Solution. Angles BAE and BCD (diagram on the right) are equal to each other, as angles with normal legs. Since $AB = CH$, we get that the right triangles $ABE$ and $CHE$ are congruent, so $AE = CE$. Triangle $ACE$ is a right triangle, and from $AE = CE$ it follows that it is isosceles, which means

![](https://cdn.mathpix.com/cropped/2024_06_05_a6c386d4b461b4a2c2ceg-2.jpg?height=310&width=264&top_left_y=261&top_left_x=1204)
that the desired angle is equal to $45^{\circ}$.