Thales theorem: Let A,B,C be points on a circle. Let AB be the Diameter of the circle. Then ACB is a right angle. Proof: Let x be angle DAC. Since DA and DC are the radius of the circle they have the same length. It follows, that DAC has the same angle as DCA. Let DBC be y. We can use the same argument to show that DCB has the angle y. Since the sum of all angles in a triangle is 180 degrees, BDC is 180-2y and ADC is 180-2x. If we add these Angles we get 180 degrees since A,D,B lie on one line. We add all angles in both triangles. 180-2x+180-2y+2x+2y=360. 180+2x+2y=360. 2x+2y=180. Now we divide by 2 on both sides: x+y=90. This completes the proof since x+y is the size of the angle ACB.

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