Most circle theorem proofs start the same way: draw the centre $O$ and join it to points on the circumference. Equal radii create isosceles triangles, and isosceles triangles create equal base angles.

From there, use angle facts you already know: angles in a triangle add to $180^{\circ }$, angles on a straight line add to $180^{\circ }$, and angles around a point add to $360^{\circ }$. This is enough to build the main circle theorems from scratch.

The key chain is: angle at the centre is twice the angle at the circumference, then same segment, then cyclic quadrilateral and tangent-chord results. Always identify the chord or arc an angle stands on, and give a reason for each step.