Draw any four points on a circle. One clean equation always connects their distances.
Take any quadrilateral whose four corners all sit on a single circle, and measure the lengths of its two diagonals and its four sides. Ptolemy claimed a precise relationship always holds between them, one exact enough to let ancient astronomers compute trigonometric values centuries before trigonometry as we now know it existed. What is that relationship?
Reveal the answer
The product of the two diagonals always equals the sum of the products of the two pairs of opposite sides. Claudius Ptolemy proved this in his 2nd-century astronomical treatise and used it to build some of the earliest known tables of chords, a forerunner of the sine function, letting him calculate angles for astronomy without any of the trigonometric machinery developed many centuries later.