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 2 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

Named alongside it

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

Circle Inscribed angle Angle Central angle Geometric mean Hypotenuse Isosceles triangle Locus Power of a point Similar triangles Tangent Thales theorem

All concepts