7. Point $P$ lies inside triangle $A B C$. It is connected to all vertices of the triangle, and perpendiculars are dropped from it to the sides, forming 6 triangles. It turns out that 4 of them are equal. Does this always mean that the triangle is isosceles?
Answer: Not necessarily.

Solution: An example of such a triangle that is not isosceles. Here, $\triangle A B C$ is a right triangle, and point $P$ is obtained by the intersection of the angle bisector of $\angle B C A$ and the perpendicular bisector of side $A C$.

![](https://cdn.mathpix.com/cropped/2024_05_06_3fd104dfae7d68abe92bg-2.jpg?height=762&width=1131&top_left_y=1886&top_left_x=451)