By H. S. M. Coxeter
Among the attractive and nontrivial theorems in geometry present in Geometry Revisited are the theorems of Ceva, Menelaus, Pappus, Desargues, Pascal, and Brianchon. a pleasant facts is given of Morley's notable theorem on attitude trisectors. The transformational viewpoint is emphasised: reflections, rotations, translations, similarities, inversions, and affine and projective ameliorations. many desirable houses of circles, triangles, quadrilaterals, and conics are constructed.
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92. The third pedal triangle is similar to the w i g i d triangle. 9B The proof is surprisingly simple. The diagram practically gives it away, as soon as we have joined P to A. Since P lies on the circurncircles of all the triangles ABlCl, A2BlC2, AdaC2, AsB~CI,and Ad2C8, we have = LBaC2P = LB8AsP and L PABl = L PCIB1 = L PCIAs = L PBsAs PEDAL POLYGONS 25 In other words, the two parts into which A P divides LA (marked in the diagram with a single arc and a double arc) have their equal counterparts at B1 and Cl, again at Ct and B2, and finally both a t Aa.
71. The angle between the Simson lines of fwo points P and P' on the circumcirck is half the angular measure of the arc P'P. If we imagine P to run steadily round the circumcircle, the line AU will rotate steadily about A at half the angular velocity in the opposite sense, so as to reverse its direction by the time P has described the whole circumference. Meanwhile, the Simson line will turn in a corresponding manner about a continuously changing center of rotation. I n fact, the Simson line envelops a beautifully symmetrical curve called a deltoid or "Steiner's hypocycloid" [20].
The center of the nine-point cirde lies otz the Euler line, midway between the orthocenter and the circumenter. 8B 22 POINTS,LINES CONNECTED WITH A TRIANGLE The history of these two theorems is somewhat confused. A problem by B. Bevan that appeared in an English journal in 1804 seems to indicate that they were known then. They are sometimes mistakenly attributed to Euler, who proved, as early as 1765, that the orthiG triangle and the medial triangle have the same circumcircle. In fact, continental writers often call the circle "the Euler circle".