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Concept

Incircle

The incircle of a triangle is the largest circle that fits inside it, touching all three sides. Its centre is where the three angle bisectors meet, and its radius is the triangle's area divided by half its perimeter.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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 Malfatti's three circles against the three largest, one at a time. A triangle with angles 40, 70, 70 degrees: Malfatti's circles cover 69.1% of it, the greedy choice of incircle then largest remaining circles covers 73.7%.

Three circles that touch and are not the largest

In 1803 Gian Francesco Malfatti asked how to cut three round columns from a triangular block of marble with as little waste as possible, and answered with three circles each touching the other two and two sides. The circles exist in every triangle and are found by solving three equations. They are never the answer to his question: the plain greedy rule — the inscribed circle first, then the largest circle that still fits, twice — always does better, by 1.4 per cent on the equilateral triangle and by almost double on a thin one.

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 Two triangulations, the same sum of cos 3x, different sums of cos 2x. Cyclic heptagon, triangulations 0 and 29: Σcos3x 1.52896873 / 1.52896873; Σcos2x -0.789261 / -0.467086.

The sums a triangulation cannot change

The Japanese theorem says the inradii of any triangulation of a cyclic polygon add to the same total. Write the inradius as a sum of cosines of the angles, and the theorem becomes one case of a rule that fits in a line: a sum of f over every angle of every triangle ignores the triangulation exactly when f(x) + f(π − x) is constant. cos 3x qualifies, cos 2x does not, and the proof is the inscribed-angle theorem applied to a single flipped diagonal.

geometry · Inscribed angle

Named alongside it

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

Catalan numbersCircumcircleInscribed angleInvariantTriangulationAngle bisectorConverseCounterexampleCyclic polygonCyclic-quadrilateralElliptic curveEuler relation

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