Home > Geometry, Uncategorized > Square inscribed in a triangle

## 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 $S$ lies inside a triangle $T$, then $2 \cdot Area(S) \leq Area(T)$.

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

Proof: 1. Denote $ABC$ the triangle and consider in the exterior of the triangle the square $BCDE$. Denote $X=AD\cap BC,Y=AE\cap BC$. Then the square of side $XY$ constructed in the same half plane as $A$ 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 $f: S \to [0,\infty)$ by $f(X)=Area(X)$ is continuous. Here we denote by $S$ the set of squares contained in the triangle $ABC$. This set is bounded and closed as a subset of $\Bbb{R}^8$ (if we think of the coordinates of the vertices). Therefore $f$ attains its maximum.

2.1.