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Concept

Circumcircle

The unique circle passing through the three corners of a triangle, centred where the three perpendicular bisectors meet. Whether it is empty of other points is the condition defining the Delaunay triangulation.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

The triangulation as the underside of a hull. Eight points on a floor, lifted onto a paraboloid above them. The faces of the lifted set's lower convex hull are shaded, and their shadows on the floor are the Delaunay triangulation of the original points.

One dimension up, and the circles disappear

The Delaunay triangulation is defined by a condition about circles, which is awkward to compute and awkward to reason about. Lift every point onto a paraboloid and the circles turn into planes, the condition turns into convexity, and a two-dimensional problem is solved by looking at a three-dimensional shape from underneath.

geometry · Voronoi The shortest tree was already in the triangulation. The Delaunay triangulation of 20 sites in faint lines with the minimum spanning tree drawn over it in heavy ones. Every tree edge is a triangulation edge, and the circles on the longest few tree edges as diameters contain no other site.

The tree inside the triangulation

The shortest network joining a set of points is built from edges chosen by length, and the triangulation is built from edges chosen by an emptiness condition about circles. The two constructions share no step, and every edge of the first is an edge of the second.

geometry · Voronoi Each side over the sine of the opposite angle is the diameter. A triangle inscribed in a circle, with the diameter from one corner drawn and joined to a second corner to make a right-angled triangle; the angle opposite the side at the far end of the diameter equals the triangle's own angle opposite that side.

Every side measured by one diameter

In any triangle, divide each side by the sine of the angle opposite it: the three answers are equal. That much is the law of sines, and it is usually left there. The common answer has a name — it is the diameter of the circle through the triangle's corners — and the reason is the inscribed angle once more. It is also why the sine was first a half-chord, why Ptolemy's theorem is the addition formula, and why a circle can hold infinitely many points all at rational distances from each other.

geometry · Inscribed angle Three lines through a point, reflected in the bisectors, meet again. A triangle with a point P and its cevians, their reflections in the angle bisectors, and the point P* where the reflections meet — P's isogonal conjugate, at (373.8, 290.0).

Every point has a partner across the bisectors

Draw the three lines from the corners of a triangle through any point, reflect each in the bisector of its own angle, and the three reflections meet again. The pairing this makes swaps the centroid with the symmedian point and the orthocentre with the circumcentre, bends every straight line into a conic through the corners, and sends the circumcircle to infinity.

geometry · Triangle centres Five triangles between the same two circles, every one closing. An outer circle of radius 1 and an inner of radius 0.38 at Euler's distance 0.4899; five triangles inscribed in the first and circumscribed about the second, started from different points.

A triangle that fits once fits everywhere

Put one circle inside another and try to fit a triangle between them, its corners on the outer circle and its sides touching the inner. Usually no triangle fits. But if one does, then one fits starting from every point of the outer circle — and whether it does is decided by a single equation in the two radii and the distance between the centres.

geometry · Triangle centres 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.

Delaunay triangulationIncircleInscribed angleVoronoi diagramAngle bisectorBarycentric coordinatesCatalan numbersChordComplexityConicConverseConvex hull

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