Triangles¶
Recall (doc/postdoc). Not a first course.
Job: \(2\)-simplex metric geometry; congruence criteria; spherical excess later for curvature.
Core¶
\[
\text{sides }a,b,c\text{ opposite }A,B,C;\quad A+B+C=\pi\ \text{(Euclidean)}
\]
| Law | Formula |
|---|---|
| Sines | \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R\) |
| Cosines | \(c^2=a^2+b^2-2ab\cos C\) |
| Area | \(\frac12ab\sin C\); Heron \(\sqrt{s(s-a)(s-b)(s-c)}\) |
| Pythagoras | \(c^2=a^2+b^2\) iff \(C=\pi/2\) |
Congruence: SSS, SAS, ASA, AAS, HL (right). SSA ambiguous (not always).
Inequality: \(a+b>c\) etc. (strict triangle inequality).
Used later for¶
- Congruence & similarity
- Trigonometry
- Differential geometry: geodesic triangles, curvature
- Complex analysis: triangle inequalities in \(\mathbb{C}\)
Check¶
Why does SSA fail as a congruence criterion in Euclidean geometry?
Given \(a,b,A\) with \(A\) acute and \(a<b\sin A\) impossible; \(h<a<b\) can yield two distinct triangles (ambiguous case).