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Miloslav Ciz 2023-12-26 14:46:41 +01:00
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triangle.md
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@ -29,7 +29,7 @@ Some basic facts, features and equations regarding triangles are following (bewa
- **Triangle angles add up to 180 degrees** ([pi](pi.md) [radians](radian.md)). This can be used to determine unknown side angles.
- Center of weight: average the three coordinates, or take the intersection of the triangle's medians.
- [Area](area.md):
- **[area](area.md)**:
- general triangle: *a * altitude(a) / 2*
- right triangle: *a * b / 2*
- **[Pythagorean theorem](pythagorean_theorem.md)**: For the lengths of the sides of a RIGHT triangle it always holds that *a^2 + b^2 = c^2*. This is extremely important and can be used to determine unknown side lengths of right triangles.
@ -38,7 +38,9 @@ Some basic facts, features and equations regarding triangles are following (bewa
- **Law of sines**: *a / sin(alpha) = b / sin(beta) = c / sin(gamma)*
- **Law of cosines**: Generalization of Pythagorean theorem: *a^2 = b^2 + c^2 - 2 * b * c * cos(alpha)*.
- Triangle [tessellation](tessellation.md) is one of only three possible regular plane tilings (the other two being [square](square.md) and [hexagon](hexagon.md)).
- Every triangle has one [incircle](incircle.md) ([circle](circle.md) inside the triangle which touches each of its sides at one point) and one [circumcircle](circumcircle.md) ("outside" circle passing through all three triangle's vertices).
- Every triangle has two special associated [circles](circle.md):
- **[incircle](incircle.md)**: circle inside the triangle which touches each of its sides at one point, its center (incenter) lies on the intersection of all angle bisectors.
- **[circumcircle](circumcircle.md)**: circle outside the triangle which touches each of its vertices, its center (circumcenter) lies on the perpendicular bisectors of each side.
- Triangle vertices always line in a single [plane](plane.md) (unlike other polygons).
In non [Euclidean](euclidean.md) ("crazy") geometries triangles behave weird, for example we can draw a triangle with three right angles on a surface of a [sphere](sphere.md) (i.e. its angles add to more than 180 degrees). This fact can be exploited by inhabitants of a space (e.g. our [Universe](universe.md)) to find out if they in fact live in a non Euclidean space (and possibly determine the space's exact [curvature](curvature.md)).