Question

In two concentric circles, chord AB of the outer circle cuts the inner circle at C and D. Prove that AC = BD.

Answer 1

Chord AB of the outer circle cuts the inner circle at C and D.
To prove: AC = BD
Construction: Draw OM = AB
Proof : Since $OM\perp AB$ (by construction)
OM also $\perp$ CD (ACDB is a line)
In the outer circle
AM = BM (1) ($\because$ OM bisects the chord AB)
In the inner circle CM = DM (2) ($\because$ OM bisects the chord CD)
From (1) and (2), we get
AM - CM = BM - DM
AC = BD