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.


Categories: Geometry, Uncategorized Tags: ,
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