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Concept

Angle bisector

An angle bisector is the line that splits an angle into two equal halves. Every point on it is equally far from the two sides of the angle, which is why a circle touching both sides has its centre there, and why the three bisectors of a triangle meet.

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

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 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

Named alongside it

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

Barycentric coordinatesCircumcircleConicCounterexampleGreedy algorithmIncircleIsogonal conjugateLine at infinityNewtons methodOptimisationReflectionSymmedian

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