If two straight lines in a circle cut one another then the rectangle contained by the pieces of one is equal to the rectangle contained by the pieces of the other. * For let the two straight lines $AC$ and $BD$, in the circle $ABCD$, cut one another at point $E$. * I say that the rectangle contained by $AE$ and $EC$ is equal to the rectangle contained by $DE$ and $EB$.
If in a circle two chords $\overline{AC},$ $\overline{BD}$ intersect at the point $E$ and let the lengths of segments be defined by $|\overline{AE}|$, $|\overline{EC}|$, $|\overline{DE}|$, and $|\overline{EB}|.$ Then the respective rectangles built from these segments have equal areas, i.e. $$|\overline{AE}|\cdot|\overline{EC}|=|\overline{DE}\cdot|\overline{EB}|.$$
Proofs: 1
Sections: 1