On a Chain of Circle Theorems

Abstract

/ / P l t P2, P3, P* are four points on a circle C, and P234 is the orthocentre of triangle P2 P 3 P4. Piu the orthocentre of triangle P1P3 P4 and so on, then the quadrilateral P234 PISA -P124 -P123 t s congruent to the quadrilateral PlP2PsPi. This theorem seems to be due to Steiner (Oes. Werke, 1, p. 128; see H. F. Baker, Introduction to Plane Geometry, 1943, p. 332) and has appeared frequently since in collections of riders on the elementary circle theorems. It is clear that P234 P134 P124 P123 lie on a circle C1234 equal to the original circle C. But also angle P 3 P134 P4 = P4 Pj P 3 = P4 P2 P 3 = PsP23iPi (with angles directed and equations modulo 77), and hence P 3 P4 P134 P234 lie on a circle C3i equal to C, and which is in fact the mirror image of C in P3P4. Similarly we obtain circles C12, C13, C14, C23, C24, so that we have in all eight circles with four points on each. If any one of these be taken as the original circle, the same system of eight circles is obtained ; if, e.g., we begin with PsPiP^ P234 on the circle C34, the four orthocentres are Pl 5 P2, P123, P m lying on C]2 and the remaining circles are the images of C34 in the six sides of the quadrangle P3 P4 P134 P234. Call this configuration KA. Let us now take a fifth point P5 on C. Then any four of Pj P2 P3 P4 P5 give a Ki. We have in fact five points Pa . . . . P5, ten points P123 . . . . P345, a circle C, ten circles C12 . . . . C45 and five circles C1234 . . . . C2345. Then the circles Cl2U C1235 C12i5 C13ii C2345 all pass through a point Pi2u&, completing a system of 16 poinis and 16 circles, five points on each circle and five circles through each point. We may show this by taking the circle C12, e.g., on which lie the five points P 1 P 2 P ) 2 3 Pi 2 4 Pi 2 5 and build up the K^s obtained by taking these four at a time. Use a parallel notation and write Q1 = Plt Q2 = P2,