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