ExplorationTake an equilateral triangle. Pick any point P inside the triangle.

Construct the segments from P to each of the sides along perpendiculars to each of the three sides.

Construct an

altitudeof the equilateral triangle.

Compare the measures of the sum of the three segments from P and the measure of the altitude. Move P to different locations.

Click HERE for a GSP 4.01 file for these explorations.

The following theorem is displayed:

Theorem:For any point P within an equilateral triangle, the sum of the perpendiculars to the three sides is equal to the altitude of the triangle.

This theorem is known as VIVIANI'S theorem. Who was Viviani? (1602-1703)

Find or create at least four different proofs of this theorem and critique each for use with students.

## EXTENSIONS

1. Parallelograms

2. Regular polygons

3. Any equiangular polygon

4. Regular polyhedra (no converse)

5. External points