chord multiplication property (idea)

States that, for any two chords in a circle (AB and CD) that intersect at point X, AX×BX = CX×DX.

Proof:

1. Angles AXC and DXB are congruent (by vertical angle theorem)
2. Construct line segments AC and DB (by line postulate)
3. Angles BAC and CDB are congruent (because they both intercept arc DB)
4. Triangles AXC and DXB are similar (by AA postulate)
5. AX/DX = CX/BX (by CLSFP postulate)
6. AX×BX = CX×DX (by multiplication).

Q.E.D.