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Concept

Projective geometry

The geometry of incidence, in which parallel lines meet at a point at infinity and only properties preserved by projection from a point are kept. Cross-ratios survive and lengths do not, so its theorems are about lines meeting and points lining up.

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

The circle is used once, and its centre is the point. A circle with its centre and one diameter, a point above it, and the straightedge-only construction of the parallel to that diameter through the point.

One circle, and a straightedge

A straightedge alone cannot bisect a segment, so it cannot draw a parallel, so it can construct almost nothing. Draw one circle anywhere and mark its centre and everything a compass could ever have done becomes available — the circle is never needed again.

computation · Compass-only The butterfly: wings that cut a chord equally. A circle with chord PQ, its midpoint M, two chords through M, and the wings AD and BC meeting PQ at X and Y, each 0.476 from M.

Wings that cut a chord equally

Take a chord of a circle and its midpoint. Draw any two more chords through the midpoint, join their ends crosswise, and the two crossing lines — the butterfly's wings — cut the first chord at equal distances from the middle. The proof is the inscribed angle and the power of a point working together; move the point off the middle and what survives is a law about reciprocals; replace the circle by any conic and the theorem does not notice.

geometry · Inscribed angle

Named alongside it

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

CircleChordConicConstructible numberConstructionHarmonic conjugateInscribed angleInvariantMidpointParallelPower of a pointSimilar triangles

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