This is a STILL image.

It is NOT a movie.

Euclid's Elements, Book 3, Proposition 35

Heath (1908) states the theorem like this: If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.

In the diagram above, this means AF.AC = AG.AB.

Interestingly, in the Elements, the theorem is not proved using similar triangles, as is common now, as they are are not dealt with systematically until Book 6 (Application of the theory of proportion). Instead, a more complex and rather opaque proof is given, involving areas of rectangles. However, the rectangles are not drawn so that the proof has a strong algebraic feel for the modern reader.

The proof rests on Book 2, Proposition 5: If a line (AB) be divided into two equal parts (at C), and also into two unequal parts (at D), the rectangle (AD.DB) contained by the unequal parts, together with the square on the part (CD) between the points of section, is equal to the square on half the line. (Casey, 1885)

This proposition is proved geometrically, using this elegant diagram (right).