The following theorem about circles (which can be traced back at least as far as Archimedes) does not seem to be widely known and may be used to “pep-up” a discussion of circle theorems.

THEOREM. Let two chords of a circle meet at P and make a fixed angle θ (radians, say) with each other. Then the sum of the (opposite) arc lengths, L + L′, is independent of the position of P inside the circle (and thus, taking P at the centre, this sum equals 2rθ, where r is the radius of the circle).