## Square inscribed in a triangle

1. Prove that given any triangle, we can inscribe a square in it.

2.0 Prove that there exists a square with maximum area inscribed in a given triangle.

2.1 Prove that if a square which lies inside a triangle has maximum area then two of its vertices lie on the same edge of the triangle.

2.2 Prove that if a square lies inside a triangle , then .

3. Now, we have a square inscribed in a triangle . Prove that the incenter of lies inside .

**Proof: **1. Denote the triangle and consider in the exterior of the triangle the square . Denote . Then the square of side constructed in the same half plane as will be inscribed in the triangle. This is a consequence of the similarity theorem, or of the fact that a homothety maps a square onto a square.

2.0. The function by is continuous. Here we denote by the set of squares contained in the triangle . This set is bounded and closed as a subset of (if we think of the coordinates of the vertices). Therefore attains its maximum.

2.1.