About Circle Geometry Proofs: Prove that if two secants to a circle from an external point are equal, then the intercepted chords are equal. (Please write like a column proof-statements and reasons)

Answer 1

Hi Wine M.,

Let secant P_A_B extend from the external point P and intersect the circle at A and B to form an external secant P_A and a chord A_B.

Let secant P_C_D extend from the external point P and intersect the circle at C and D to form an external secant P_C and a chord C_D.

Then from the Intersecting Secant Theorem PA*PB = PC*PD and if PA = PC then:

PA*PB = PC*PD

PC*PB = PC*PD, (substitute PA = PC)

PB = PD, (PC's cancel)

By definition, PB = PA + AB and PD = PC + CD:

PA + AB = PC + CD, (substitute for PB = PD)

PC + AB = PC + CD, (substitute PA = PC)

AB = CD

I hope this helps, Joe.