An elementary treatise on modern pure geometry (1893)
Lachlan, Robert · London, New York, Macmillan and Co. · leaf 292
r fixed points on a circle subtend a pencil, whosecross ratio is constant, at all points on the circle. If A, B, C, D be four fixed points on a circle, the pencilP {AB, CD] has a constant cross ratio for all positions of the pointP on the circle since the angles APB, APO, &c. are of… … P A B c^ D^ Let 0 be the centre of the circle. Then, since the anglesPAO, PBO are respectively complementary to half the anglesAOP, BOP, the angle AOE is equal to half the angle AOB, andis therefore equal to the angle APB.Hence we have, P {AB, CD] = 0 {AB, CD]. Hence, since the cross ratio of the…
Geometry
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