a cyclic quadrilateral

Four points scattered on a circle's rim, joined in order, make a four-sided figure whose every corner sits exactly on the circle — like four pins on a circular cork-board linked by string. Such a quadrilateral, one that can be drawn inside a single circle with all four vertices on it, is a cyclic (or inscribed) quadrilateral.

Precisely, a cyclic quadrilateral is a quadrilateral all four of whose vertices lie on one common circle (its circumcircle). Its signature property comes straight from the inscribed angle theorem: opposite angles are supplementary — each pair of opposite angles sums to 180 degrees. The reason is that the two opposite angles stand on the two arcs that together make the whole circle (360 degrees of arc), and each inscribed angle is half its arc, so together they are half of 360, namely 180. The converse is a powerful test: if a quadrilateral's opposite angles sum to 180 degrees, then it is cyclic and a circle passes through all four vertices.

Cyclic quadrilaterals are central to competition geometry and to the theory of inscribed figures: every rectangle and every isosceles trapezoid is cyclic, while a non-rectangular parallelogram is not (its opposite angles are equal, not supplementary — and equal angles sum to 180 only if each is 90). The honest caveat: not every quadrilateral is cyclic. A generic four points need not lie on a common circle; three points always determine a circle, but the fourth must land exactly on it. So 'cyclic' is a special, checkable condition, captured by the supplementary-opposite-angles test (or equivalently by Ptolemy's relation among the sides and diagonals).