Q:

A chord PQ cuts a coincentric circle at P'Q', prove that PP'=QQ'

The question is from the book challenge and thrills in pre college mathematics.  
A chord PQ cuts a coincentric circle at P'Q', prove that PP'=QQ'

A:

Drop a perpendicular line from the common center of the circles to the chord PQ and call the intersection of the chord and the line R. Draw auxiliary lines from the center to points P, P', R, Q', and Q. Since radii of a circle are equal, you can easily prove using congruent triangles that RP=RP' and RQ=RQ'.  Subtracting, you get PP'=QQ'.