Proofs about circles
A circle gives a proof almost one fact to start from, that every radius has the same length. How far does that one fact go?
Where you are 3 inside
What this area is about
Every point of a circle is the same distance r from its center. Proofs about circles start from little else, and it goes a long way: two radii make an isosceles triangle. This area proves circle facts you may only have measured, so each can be cited in a two-column proof.
How these fit together
The base. Proving the inscribed angle theorem shows that an inscribed angle measures half the arc it intercepts, m\angle APB = \tfrac{1}{2}\, m\overset{\frown}{AB}. Its first case is the isosceles triangle made by two radii. The same lesson cites it twice to prove that a cyclic quadrilateral’s opposite angles add to 180^\circ, and proves the converse.
Built on it. Proving the power of a point relationships uses the theorem for equal angles; similar triangles give equal products such as PA \cdot PB = PC \cdot PD.
Apart. Proving a tangent is perpendicular to the radius needs no arcs: an indirect proof shows the radius to the point of contact is perpendicular to the tangent.
A path through it
The inscribed angle theorem first, the tangent theorem any time before power of a point. Power of a point goes last. It also needs similar triangle proofs, the tangent theorem for PT^2 = OP^2 - r^2, where \overline{PT} is a tangent from an outside point P and O the center, and the cyclic quadrilateral converse for its own converse.
No later star builds on these proofs yet. They back facts that cyclic quadrilaterals and tangent and radius state and use.