Intersecting chords theorem

Statement

Let $A$, $B$, $C$ and $D$ be concyclic points of the Euclidean plane $\mathbb{R}^2$, and let $d$ be the Euclidean distance. Suppose that the chords $AC$ and $BD$ meet at a point $S$ that is inside both segments. Then $$d(A, S)\, d(S, C) = d(B, S)\, d(S, D).$$

Remarks

Both products are equal to $r^2 - d(S, O)^2$, where $O$ is the centre and $r$ the radius. This is minus the power of $S$. A proof: the triangles $ASD$ and $BSC$ are similar, by the inscribed angle theorem. The theorem is Proposition 35 of Book III of Euclid's Elements.

Explanations

L1 human

Draw two chords of a circle that cross each other. Each chord is cut into two pieces. The product of the two pieces is the same for both chords.