Two chords AB and

Given: AB and CD are chords of a circle intersecting at point P outside the circle.

(a). To Prove: ∆PAC ∆PDB

Proof: We know

ABD + ACD = 180° [ opposite angles of cyclic quadrilateral are supplementary] …(i)

PCA + ACD = 180° [ they are linear pair angle] …(ii)

Comparing equations (i) & (ii), we get

ABD + ACD = PCA + ACD

ABD = PCA

Also, APC = BPD [ they are common angles]

Thus, by AA-similarity criteria, ∆PAC ∆PDB

Hence, proved.

(b). To Prove: PA × PB = PC × PD

Proof: We have already proved that, ∆PAC ∆PDB

Thus the ratios can be written as,

By cross-multiplication, we get

PA × PB = PC × PD

Hence, proved.

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