8,9 |

In an acute triangle $A B C$, altitudes $A L$ and $B M$ are drawn. Then, the line $L M$ is extended to intersect the extension of side $A B$.

What is the maximum number of pairs of similar triangles that can be counted on this diagram, if no pairs of congruent triangles are formed on it?
Points $A, B, M$ and $L$ lie on the same circle.

## Solution

Right triangles $C A L$ and $C B L$ are similar by two angles.

Let $H$ be the orthocenter of triangle $A B C$. Right triangles $A M H$ and $B L H$ are similar by two angles.

Right triangles $A M H$ and $A L C$ are similar by two angles.

Right triangles $B L H$ and $B M C$ are similar by two angles.

Therefore, right triangles $A M H$ and $B M C$ are similar, as well as right triangles $B L H$ and $A L C$.

Points $M$ and $L$ lie on the circle with diameter $A B$. Therefore, triangles $M H L$ and $A H B$ are similar by two angles.

Triangles $C L M$ and $C A B$ are also similar.

Let the lines $A B$ and $M L$ intersect at point $D$. Similarly, triangles $D B L$ and $D M A$ are similar by two angles.

Finally, from the equality $\angle B M L = \angle B A L$, it follows that triangles $D B M$ and $D L A$ are similar.

![](https://cdn.mathpix.com/cropped/2024_05_06_7adbd1f2e5e559c8684fg-48.jpg?height=306&width=520&top_left_y=224&top_left_x=769)

Answer

10.