At least Three Different Problems

Consider any triangle ABC. Construct a square to the outside of the triangle on each side.

Generate a GSP construction to explore this configuration.

Select any two square centers and the midpoint of the other side of triangle ABC. Explore the triangle formed by these three points.

Conjecture?

Proof?

Explore the relationship of two segments, one being defined by a vertex of the triangle and the center of the square on the opposite side and the other being defined by the centers of the squares on the sides adjacent to the vertex.

Conjecture?

Proof?

Note that there are three pairs of such segments.

Explore the same relationships when the square on each side is constructed toward the interior of the triangle

Prove that the three lines from the vertices to the centers of the squares on the opposite sides are

concurrent.

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