In the following figure; P and Q are the points of intersection of two circles with centres O and O’. If straight lines APB and CQD are parallel to O O'; prove that:
(i) O O' = `1/2AB` (ii) AB = CD
Drop OM and O'N perpendicular on AB and OM' and O'N' perpendicular on CD.
∴ `MP = 1/2AP,PN=1/2 BP, M'Q =1/2CQ, QN'=1/2QD`
Now,` OO ' = MN = MP + PN = 1/2 (AP + BP) = 1/2 AB` ---------(i)
And `OO ' = M ' N ' = M 'Q + QN ' = 1/2 (CQ + QD) = 1/2 CD` -------(ii)
By (i) and (ii)
AB = CD