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Miloslav Ciz 2024-11-19 22:31:10 +01:00
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triangle.md
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@ -4,7 +4,7 @@ Triangle is a three sided [polygon](polygon.md), one of the most basic [geometri
{ In my favorite book [Flatland](flatland.md) triangles represent the lowest class of men with isoscele triangles being the lowest as they are most similar to [women](woman.md) who are just straight [lines](line.md). ~drummyfish }
Triangle consists of three [vertices](vertex.md) (usually labeled *A*, *B* and *C*), three sides (usually labeled *a*, *b* and *c* according to the side's opposing vertex) and three angles (usually labeled *alpha*, *beta* and *gamma* according to the closest vertex):
Triangle consists of three [vertices](vertex.md) (usually labeled *A*, *B* and *C*), three sides (usually labeled *a*, *b* and *c* according to the side's opposing vertex) and three angles (usually labeled *alpha*, *beta* and *gamma* according to the closest vertex, but may also just be labeled *A*, *B* and *C*):
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@ -37,11 +37,13 @@ Some basic facts, features and equations regarding triangles are following (bewa
- **Triangle inequality**: Sum of any two side lengths can't be greater than the length of the third side, i.e. *a + b <= c*. That means that e.g. a triangle with side lengths 1, 2 and 4 can't exist because 1 + 4 > 2. If one side of a triangle is exactly the sum of the other two, the triangle is called **degenerate**, its vertices lie on the same line and it is completely "squashed".
- **Law of sines**: *a / sin(alpha) = b / sin(beta) = c / sin(gamma)*
- **Law of cosines** (generalization of Pythagorean theorem to any triangle, not just right one): *a^2 = b^2 + c^2 - 2 * b * c * cos(alpha)*.
- The greatest angle is opposite to the greatest side.
- 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 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) -- it's pretty clear but good to realize e.g. in 3D graphics, every face in a triangle mesh will always have a clearly defined normal etc.
- ...
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)).