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Cavalieri's principle in 2D

Example 5. The area of the median triangle is 3/4 of the area of the original triangle. The solution is based on the fact established in Example 4 and on two transformations. First we use a translation of two of the medians to build the median triangle. The second transformation is shearing (also known as the 2D case of Cavalieri's principle) of the three triangles forming the median triangle. More details.
Example 6.  The Law of Cosines Like in Example 3, the animation in this applet creates a good visualization of the proof, but in order for the proof to be completed it needs to be supplemented by a couple of equations. See f.[4] More details.
Example 7. Pythagorean Theorem The proof of the Pythagorean Theorem follows from the Law of Cosines. More details.