Circle Theorems¶
Recall (doc/postdoc). Not a first course.
Job: classical Euclidean constraints on angles/chords; power as \(PT^2=PA\cdot PB\) = absolute value of circle polynomial.
Core¶
| Theorem | Statement |
|---|---|
| Inscribed angle | angle at circumference \(= \frac12\) central subtending same arc |
| Semicircle (Thales) | angle in semicircle is right |
| Same arc | inscribed angles subtending same arc equal |
| Cyclic quad | opposite angles sum to \(\pi\) |
| Tangent-chord | angle between tangent and chord \(=\) angle in alternate segment |
| Power / intersecting chords | \(AE\cdot EB=CE\cdot ED\) (internal); secants \(PA\cdot PB=PC\cdot PD\) |
| Tangent-secant | \(PT^2=PA\cdot PB\) |
Power of \(P\) w.r.t. circle center \(O\) radius \(r\): \(|OP|^2-r^2\) (signed).
Used later for¶
- Complex analysis: argument principle geometry
- Inversion geometry: power, radical axis
- Algebraic geometry: circle as quadric
- Triangles: circumcircle, \(a/\sin A=2R\)
Check¶
Does Thales' theorem require the Euclidean parallel postulate?
Yes in the sense that “angle in semicircle is right” characterizes Euclidean geometry among neutral geometries (equivalent forms of Euclid V).