# Triangle

**Type:** Concept  
**Domain:** Geometry  
**Codex URL:** /mathematics/geometry/triangle/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
The simplest closed polygon. Rigid shape (basis of structural engineering). Foundation of trigonometry.

## Congruence criteria
SSS, SAS, ASA, AAS determine congruence. AAA only gives similarity, not congruence.

## Four special points
Centroid, circumcentre, incentre, orthocentre. Centroid+circumcentre+orthocentre lie on the Euler line (1765).

## Law of Sines & Cosines
a/sin(A)=b/sin(B)=c/sin(C); c²=a²+b²−2ab·cos(C)

## Heron's formula
Area=√[s(s−a)(s−b)(s−c)], s=semi-perimeter. Special case (d=0) of Brahmagupta's cyclic quadrilateral formula.

## Nine-point circle
9 special points (midpoints, altitude feet, Euler-point midpoints) all lie on one circle (Feuerbach, 1822).

## Curved-space triangles
Sphere: angle sum > 180° (proportional to area). Hyperbolic: angle sum < 180°. Direct evidence of curvature.

## Sources
### Tier 1
- Coxeter, H.S.M. and Greitzer, S.L. (1967). *Geometry Revisited*.
- Hartshorne, R. (2000). *Geometry: Euclid and Beyond*.
### Tier 2
- Johnson, R.A. (1929/2007). *Advanced Euclidean Geometry*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*