Let the chords PQ and RS of a circle intersect at point X. We need to prove that the product of the segments of one chord equals the product of the segments of the other chord, i.e., \[ PX \cdot XQ = RX \cdot XS \] Proof: In triangles ∆PXS and ∆RXQ: ∠SPQ = ∠SRQ [They subtend the same arc of the circle] ⇒ ∠SPX = ∠XRQ Also, ∠PSR = ∠PQR [They subtend the same arc] ⇒ ∠PSX = ∠RQX And ∠PXS = vertically opposite ∠RXQ ∴ ∆PXS and ∆RXQ are similar triangles. So, \[ \frac{PX}{RX} = \frac{XS}{XQ} = \frac{PS}{RQ} \] That implies, \[ \frac{PX}{RX} = \frac{XS}{XQ} \] Or, \[ PX \cdot XQ = RX \cdot XS \quad \text{(Proved)} \]