Matpic
mathematics, in pictures
Concept

Similar triangles

Triangles with the same angles, so that one is an enlargement of the other and their sides are in fixed ratio. The fixed ratio of their sides is what makes trigonometry possible, since the ratios depend on the angles and not on the size.

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

Euclid's proof, without moving anything. The square on a leg and its share of the square on the hypotenuse are each exactly twice the same triangle, so they are equal. Nothing in the figure is cut or rearranged; the triangle is only looked at from the other side.

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.

geometry · pythagoras 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 Inversion in a circle of radius 1. Three points and their images under inversion in a circle: each image lies on the same ray from the centre, at the distance whose product with the original is the squared radius. Beside it, the tangent construction that finds the image with compass and straightedge.

The map that trades circles for lines

Send every point to the one on the same ray whose distance multiplies with it to a fixed number, and circles become lines, lines become circles, angles survive untouched, and a ring of tangent circles falls out of a ring of equal ones.

geometry · inversion

Named alongside it

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

AltitudeCircleAreaCircle preservingConformal mapCongruenceConstructionCounterexampleDissectionHypotenuseIncidenceInscribed angle

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