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Challenge 328: Top Trapezium

The circumferences of circles ­C1 and C2 touch at exactly one point.

Both circles also touch a straight line L, at points Q1 and Q2 respectively.

The centre of C1 is P1, and the centre of C2 is P2.

The radius of C1 is R and the radius of C2 is r.

(i) Find the area of the trapezium T with vertices Q1P1P2Q2 in terms of R and r

(ii) Given that R + r = 10 units, find the maximum area of T. Prove that your answer really is the maximum value.