chord AB and CD when produced meet at point P.

Question

If ∠AOC=θ,∠BOD=α
then prove that ∠APC= (θ−α)/2

( O is the centre of the circle )

Answer 1

Consider the extremes to get a feel for this one.
If the two chords are parallel, then the lines never meet (or meet infinitely far away, so α = θ, and angle P = 0

If P lies on the circle, then the arc BD is zero and thus α = 0, so angle P is an inscribed angle, and is α = θ/2

Let AC be a diameter and CD be a tangent line. In that case, CD = 0, and θ = 180, so α = 0 and angle P is 180/2 = 90

So the extreme cases all work. So now look up theorems on extended secants and major/minor arcs. It should not take long to come up with a proof.