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 9 essays across 2 fields — 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 A compass that will not change its opening. A segment longer than twice the compass's fixed opening, with the opening stepped along it 2 times and the remaining piece bisected by two arcs of that same opening.

The compass that will not open

Fix the compass at one opening and never change it. That looks like a serious loss — a circle of a given radius through a given point is the compass's whole job — and it turns out to cost nothing at all, for reasons that are arithmetic rather than geometric.

computation · Compass-only The nine-point circle, touching four others. A triangle with its nine-point circle, its inscribed circle and its three escribed circles, each of the four tangent to the first — with the distances between centres compared against the radii.

One circle touching four

The nine-point circle touches the inscribed circle and each of the three escribed ones. Nothing in its construction mentions them, the two families of centres are built from different kinds of number, and the tangency is four exact equalities between distances and radii.

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 Every chord through the point cuts into pieces whose product is 16.00. A circle with a point inside it and four chords drawn through the point, each labelled with the lengths of its two pieces, beside four rectangles whose sides are those pieces and whose areas are all equal.

One number for every chord through a point

Draw any line through a point and let it cut a circle twice. The two distances from the point to the circle multiply to the same number whichever line is drawn — inside, outside, or grazing as a tangent. The number belongs to the point, and the reason it does not depend on the line is the inscribed angle: two chords through a point cut out two triangles with the same angles.

geometry · Inscribed angle Bands of a sphere and of its cylinder, cut by the same planes. A unit sphere inside its cylinder with 3 horizontal bands; each sphere band has area 1.885, 1.885, 1.885, equal to the cylinder band of the same height.

A band of a sphere is a band of its cylinder

Cut a sphere with two parallel planes and the band between them has exactly the area of the band the same planes cut from the cylinder wrapped round the sphere — whatever the band's latitude. Near the equator the sphere's band is wide and nearly upright; near a pole it is narrow and nearly flat; and the two effects cancel exactly. Archimedes proved it, wanted it on his tomb, and it gives the area of the sphere, the only honest way to pick a random point on it, and a map on which no country is the wrong size.

geometry · Circle area 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.

CircleInscribed angleInvariantAltitudeCounterexampleIncidenceLocusAreaConstructionPerpendicular bisectorPower of a pointSymmetry

All concepts