Concept

Triangle centres

Points defined from a triangle so that they do not depend on how it is placed or labelled, such as the centroid, the incentre and the circumcentre. Thousands are catalogued, and the relations between them are a large part of triangle geometry.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Trisect every angle, and an equilateral triangle appears. A triangle with angles 78°, 54°, 48°, its six angle trisectors, and the triangle whose corners are where the trisectors nearest each side meet. That inner triangle is equilateral, which is Morley's theorem.

Three trisectors and a triangle nobody expected

Cut every angle of a triangle into three. The trisectors nearest each side meet in three points, and those three points are always the corners of an equilateral triangle — for every triangle there is, with no exceptions and no reason anybody finds obvious.

geometry · Morley

Two ways to cut a hexagon, one total of radii. A cyclic hexagon triangulated two ways with the incircles drawn; the inradii sum to 1.1622 both times.

A total hung in a temple

Put a polygon's corners on a circle, cut it into triangles, and add up the radii of the circles inscribed in the triangles. Cut it a different way and the total is the same — for all fourteen ways of cutting a hexagon, to every digit. The fact was painted on a wooden tablet in a Japanese temple around 1800, and the reason for it is a theorem about one triangle and the distances from its circumcentre to its sides.

geometry · Inscribed angle

Named alongside it

The objects these essays reach for when they reach for this one.

Exhaustive search Angle Catalan numbers Circumcircle Construction Converse Counterexample Cyclic polygon Equilateral triangle Incircle Inscribed angle Invariance

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