The "intersecting chord segment product property" a.k.a. the "intersecting chords theorem" says the products of the two segments of chords cut by their point of intersection are equal. That is,
(BP)(CP) = (DP)(AP)
To see why this is true, note that triangles ABP and CDP are similar. From the Central Angle Theorem, we see that angles A and C are the same, and for the same reason, angles B and D are the same. The angle P of each triangle is the same by Vertical Angles, so the triangles are similar. Setting corresponding sides in proportion,
BP / DP = AP / CP,
and the result follows.
Summary of geometrical theorems
The other Intersecting Chords Theorem is a generalization of the central angle theorem.
Cut-the-knot: Intersecting Chords Theorem
The webmaster and author of this Math Help site is Graeme McRae.