AQ, AR, BC are tangents for the circle with centre O and BC, AD, AE are tangents to the circle with centre O\'. Prove that ABO\'C is parallelogram. It is given that radius of bigger circle is three times the smaller circle.

In triangle AOQ and AO'D the sides are in ratios as the triangles are similar

so AO/AO' = AQ/AD = OQ/O'D

On Solving we get , AO = 2r

Now , If BC is tangent at point E the AE = AO + OE = 3r = O'E

Also , BE = CE

Thus diagonals AO' and BC are bisected at point E

Hence , ABO'C is a parallelogram