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Concept

Perpendicular bisector

The line of points equally far from two given ones, at right angles to the segment joining them. The three of a triangle meet at the centre of its circumscribed circle, which is why that circle exists at all.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The plane divided by nearest neighbour. 10 sites, and every point of the rectangle shaded by which site is closest to it. The boundaries are the places where two sites tie.

The plane, divided by whoever is nearest

Scatter some points and colour every other point of the plane by which one is closest. The result is a tiling nobody designed, and its dual triangulation has a property that no part of the construction mentions.

geometry · Voronoi Nine points of a triangle, on one circle. A triangle with the midpoints of its sides, the feet of its three altitudes and the midpoints from each corner to the orthocentre marked; all nine lie on a single circle of half the circumradius.

Nine points on one circle

Three midpoints, three feet of altitudes and three more midpoints. Nine points defined in three unrelated ways, on an arbitrary triangle, and all nine sit on one circle — checked here on two hundred and forty triangles as well as on the drawn one.

geometry · Triangle centres The classical centres as three weights each. A table of triangle centres with the weights on the three corners that produce each, and the determinants that decide which triples of them are collinear.

A centre is three weights

Write each classical centre as a weighted average of the corners and a coincidence becomes a determinant. The Euler line is then one number rather than a construction, the whole catalogue becomes mechanical, and the reason one centre is missing from it is visible in the weights.

geometry · Triangle centres

Named alongside it

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

AltitudeCircleCounterexampleIncidenceInvariantLocusSimilar trianglesConvexityDelaunay triangulationDeterminantDualityEuler characteristic

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