Unit IV Summary: From Circle Invariants to the Regular Pentagon

How Books I, II & III Converge in Book IV, Propositions 10–11

Books III and IV weave together every preceding thread of the Elements:

1. Inscribed & Semicircular Angles (``III.20``, ``III.31``): Isosceles radius triangles (I.5) and the exterior-angle theorem (I.32) establish that central angles double inscribed angles and that every angle inscribed in a semicircle is right.

2. Power of a Point (``III.35``–``III.37``): Book II geometric algebra (II.5–II.6) combined with the Pythagorean theorem (I.47) proves that the square on a tangent equals the rectangle of any secant and its external part.

3. The Pentagonal Crown (``IV.10``–``IV.11``): Cutting a radius in golden ratio via II.11 and recognizing tangent contact via III.37 constructs the $36^\circ\text{–}72^\circ\text{–}72^\circ$ Golden Triangle (IV.10), whose angle bisectors immediately inscribe the regular pentagon (IV.11).