An elegant proof of pythagoras' theorem

Prove Pythagoras' theorem.

Geometer's Sketchpad

Given triangle ABC with no angle >120 degrees, find and construct the point P for which PA + PB + PC is a minimum. What is this point called? What would be the case for a triangle with an angle of 120 degrees or more? ------------------------------------------------------ I need this done in Geometer's sketchpad as well as step-by-step how the problem was solved so I can learn the process. Thanks

Heron's Formula

Illustrate Heron's Formula for finding area of triangle using Geometer's Sketchpad

Concurrency

Show that the medians of a general triangle are concurrent and that the point of concurrency divides each median in a 2:1 ratio.

Pythagorean Theorem

I need to use the theorem to show that the area on the outside of A plus the outside of B = outside of C. I can't use something as easy as a square but I don't need anything too crazy. Please help with any suggestions as well as how I would find the area.