Triangles
This file proves basic geometrical results about distances and angles
in (possibly degenerate) triangles in real inner product spaces and
Euclidean affine spaces. More specialized results, and results
developed for simplices in general rather than just for triangles, are
in separate files. Definitions and results that make sense in more
general affine spaces rather than just in the Euclidean case go under
linear_algebra.affine_space.
Implementation notes
Results in this file are generally given in a form with only those non-degeneracy conditions needed for the particular result, rather than requiring affine independence of the points of a triangle unnecessarily.
References
- https://en.wikipedia.org/wiki/Pythagorean_theorem
- https://en.wikipedia.org/wiki/Law_of_cosines
- https://en.wikipedia.org/wiki/Pons_asinorum
- https://en.wikipedia.org/wiki/Sum_of_angles_of_a_triangle
Geometrical results on triangles in real inner product spaces
This section develops some results on (possibly degenerate) triangles in real inner product spaces, where those definitions and results can most conveniently be developed in terms of vectors and then used to deduce corresponding results for Euclidean affine spaces.
Pythagorean theorem, if-and-only-if vector angle form.
Pythagorean theorem, subtracting vectors, if-and-only-if vector angle form.
Law of cosines (cosine rule), vector angle form.
Pons asinorum, vector angle form.
Converse of pons asinorum, vector angle form.
The cosine of the sum of two angles in a possibly degenerate triangle (where two given sides are nonzero), vector angle form.
The sine of the sum of two angles in a possibly degenerate triangle (where two given sides are nonzero), vector angle form.
The cosine of the sum of the angles of a possibly degenerate triangle (where two given sides are nonzero), vector angle form.
The sine of the sum of the angles of a possibly degenerate triangle (where two given sides are nonzero), vector angle form.
The sum of the angles of a possibly degenerate triangle (where the two given sides are nonzero), vector angle form.
Geometrical results on triangles in Euclidean affine spaces
This section develops some geometrical definitions and results on (possible degenerate) triangles in Euclidean affine spaces.
Pythagorean theorem, if-and-only-if angle-at-point form.
Law of cosines (cosine rule), angle-at-point form.
Pons asinorum, angle-at-point form.
Converse of pons asinorum, angle-at-point form.
The sum of the angles of a possibly degenerate triangle (where the given vertex is distinct from the others), angle-at-point.