Constant breadth

A closed, planar curve C is said to have constant breadth \mu if the distance between parallel tangent lines to C is always \mu. C needn’t be a circle. See the Wankel engine design.
a) If we call two points with parallel tangent lines opposite, prove that the chord joining opposite points is normal to the curve at both points.
b) Prove that the sum of reciprocals of the curvature at opposite points is equal to \mu.
c) (easy application) Prove (using b) ) that the circle of radius R has constant curvature \frac{1}{R}.
d) Prove that the length of C is \pi\mu.

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