The Pythagorean Theorem - Educational Outreach

However, the last four-part equivalency is not enough. Weneed information about the three angles in our arbitrarytriangle generated via the critical point x , y ) as shownin Figure 2.37. In particular, is(cp cp a right angle? If2 2 2so, then the condition A B C corresponds to the factthat LMN is a right triangle and we are done! To proceed,first rewrite x [ C x ] y2 0 ascpcpcpxycpcpycp .C xcpNow study this proportional equality in light of Figure 2.37,where one sees that it establishes direct proportionality ofnon-hypotenuse sides for the two triangles LPMand MPN . From Figure 2.37, we see that both triangles haveinterior right angles, establishing thatThus LPM MPN . and . Since the sum of the remaining0two angles in a right triangle is 90 both00 90 . Combining 90 with the equality0 90 and and the definition for immediately leads to0 90 , establishing the key fact that LMN is aright triangle and the subsequent simultaneity of the twoconditionsA2 B2 C 2Figure 2.38 on the next page summarizes thevarious logic paths applicable to the now establishedCauliflower Proof of the Pythagorean Theorem with Converse.78