Concept

Cyclic-quadrilateral

A four-sided figure whose corners all lie on one circle. Its opposite angles add to a half-turn, and Ptolemy's theorem ties its sides to its diagonals; either property is a test for whether four points are concyclic.

Named by 3 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

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

Two triangulations, the same sum of cos 3x, different sums of cos 2x. Cyclic heptagon, triangulations 0 and 29: Σcos3x 1.52896873 / 1.52896873; Σcos2x -0.789261 / -0.467086.

The sums a triangulation cannot change

The Japanese theorem says the inradii of any triangulation of a cyclic polygon add to the same total. Write the inradius as a sum of cosines of the angles, and the theorem becomes one case of a rule that fits in a line: a sum of f over every angle of every triangle ignores the triangulation exactly when f(x) + f(π − x) is constant. cos 3x qualifies, cos 2x does not, and the proof is the inscribed-angle theorem applied to a single flipped diagonal.

geometry · Inscribed angle

Named alongside it

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

Inscribed angle Circle Angle Catalan numbers Central angle Geometric mean Hypotenuse Incircle Invariant Isosceles triangle Locus Power of a point

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