Altitude
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
Also named here as similar triangles — the same set of essays touches all of them, so they are one junction rather than several.
Euclid proves it without moving anything
The rearrangement proof cuts and slides. Euclid's does neither — it shows that a square and a rectangle are each exactly twice the same triangle, seen from opposite sides, and that is harder to hold in the head for a reason worth understanding.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Similar trianglesAreaCircleCongruenceCounterexampleDissectionHypotenuseIncidenceInscribed angleInvariantLocusParallel lines