Let the obtuse angle MOQ = \(2x\). Using the circle theorem, the angle at the centre is twice the angle at the circumference. Angle MNQ = \(x\) and angle MPQ = \(x\). Therefore angle MNQ = angle MPQ.

Further material is included, for example, closed curves, enveloping curves, curves of constant width, the fundamental theorem of surface theory, constant mean curvature surfaces, and existence of ...