Matpic
mathematics, in pictures
Concept

Locus — where it appears

The set of all points satisfying a stated condition, taken as a shape in its own right. Describing a curve as one turns a geometric condition into an equation, which is the whole method of coordinate geometry.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

An angle standing on a chord. A circle with a fixed chord and a movable apex on the major arc. The angle at the apex is 60 degrees wherever the apex is put, and the angle the same chord subtends at the centre is 120 degrees.

An angle that does not care where it stands

Fix two points on a circle and look at them from anywhere else on the far arc. The angle is the same from every one of those places, and it is exactly half the angle at the centre.

geometry · Inscribed angle 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 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

Named alongside it

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

CircleCounterexampleIncidenceInscribed angleInvariantSimilar trianglesAltitudePerpendicular bisectorAngleCentral angleCyclic quadrilateralDeterminant

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