circle theorems using parallels

normally the circle theorems are justified using isosceles triangles
but it's quite neat to use alternate and corresponding angles on parallel lines

this and other ideas are explored more substantially by Dietmar Kuchemann in this article

construct a parallel line

but A = B (isosceles triangle) so the angle at the entre is double the angle at the circumference


the angle in a semicircle being 90 degrees follows from the fact that an isosceles triangle is perpendicularly bisected by its line of symmetry

construct a parallel line