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Special Invited Session: The Geometry of Triangles Abstracts

Saturday, August 8, 1:00 PM - 2:50 PM, Marriott Wardman Park, Salon 1

Richard Guy and John Conway will share their latest ideas about the geometry of Euclidean triangles.

A Triangle Has Eight Vertices (But Only One Centre)

1:00 PM - 1:50 PM
Richard GuyUniversity of Calgary

Quadration regards a triangle as an orthocentric quadrangle. Twinning is an involution between orthocentres and circumcentres. Together with variations of Conway's Extraversion, these give rise to symmetric sets of points, lines and circles. There are eight vertices, which are also both orthocentres and circumcentres. Twelve edges share six midpoints, which with six diagonal points, lie on the 50-point circle, better known as the 9-point circle. There are 32 circles, which touch three edges and also touch the 50-point circle. 32 Gergonne points, when joined to their respective touch-centres, give sets of four segments which concur in eight deLongchamp points, which, with the eight centroids, form two harmonic ranges with the ortho- and circum-centres on each of the four Euler lines. Corresponding points on the eight circumcircles generate pairs of parallel Simson-Wallace lines, each containing six feet of perpendiculars. In three symmetrical positions these coincide, with twelve feet on one line. In the three orthogonal positions they are pairs of parallel tangents to the 50-point circle, forming the Steiner Star of David. This three-symmetry is shared with the 144 Morley triangles, which are all homothetic. Time does not allow investigation of the 256 Malfatti configurations, whose 256 radpoints probably lie in fours on 64 guylines, eight through each of the eight vertices.

New Ideas about the Geometry of Triangles

2:00 PM - 2:50 PM
John ConwayPrinceton University

The geometry of triangles is an old subject. I will discuss some ideas that tie together its different parts and make it easy to remember many old and new theorems.