I have seen the following in a circle geometry proof in a Cambridge textbook:
We have proven that angles at the circumference standing on the same arc of a circle are equal. The converse of this is:
If two points lie on the same side of an interval, and the angles subtended at these points by the interval are equal, then the two points and the endpoints of the interval are concyclic.
The most satisfactory proof makes use of the forward theorem.
Let P and Q be points on the same side of an interval AB such that ∠APB = ∠AQB = α.
Aim: To prove that the points A, B, P, and Q are concyclic.
Construction: Construct the circle through A, B and P, and let the circle meet AQ (produced if necessary) at X. Join XB.
Proof: Using the forward theorem,
∠AXB = ∠AQB = α (angles on the same arc AB).
Hence ∠AXB = ∠AQB,
so QB ∥ XB (corresponding angles are equal)
But QB and XB intersect at B, and are therefore the same line.
Hence Q and X coincide, and so Q lies on the circle.
What is meant by 'the forward theorem'? Is it just referring to the original theorem of which this theorem is the converse (the 'converse of the converse')?
Or is there a theorem known as 'the forward theorem'?