Concept

Barycentric coordinates

Coordinates that locate a point by the weights it would need at a triangle's corners to balance there, normalised to add to one. They make lines, centres and ratios in a triangle linear, and they are how triangle centres are tabulated.

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

Every ear carried to a slice of a disc. The corridor's triangulation on the left and the same triangulation of a regular 20-gon on the right, with four points and their images marked; the map is affine on each triangle and agrees on every shared edge.

Every loop is a circle in disguise

Separating the plane is the weak half of what the eye believes about a closed curve. The strong half is that the inside is a disc — that the whole plane can be bent until the curve is a round circle — and for a polygon that is a construction rather than an argument.

topology · Jordan curve

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

Named alongside it

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

Angle bisector Circumcircle Closed curve Conic Continuity Counterexample Disc Homeomorphism Isogonal conjugate Jordan curve Line at infinity Piecewise linear

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