Triangle geometry/Introduction/Section

A triangle is just a tuple ${\displaystyle {}(A,B,C)}$ of three points (called vertices) in an affine space ${\displaystyle {}E}$ (typically, an affine plane) over a Euclidean space ${\displaystyle {}V}$. We allow the situation that some or all vertices coincide (a degenerate triangle), and we identify triangles if they only differ by the ordering of the vertices. A triangle is by definition not degenerate if and only if the three points are affinely independent. Quite often, we regard a triangle as its convex hull, that is, the set

${\displaystyle {\left\{rA+sB+tC\mid r+s+t=1,\,0\leq r,s,t\leq 1\right\}}}$

of all barycentric combinations of the three points, where all coefficients are not negative. The connecting segment

${\displaystyle {}{\overline {A,B}}={\left\{rA+sB\mid r+s=1,\,0\leq r,s\leq 1\right\}}\,}$

is called the edge between the vertices ${\displaystyle {}A}$ and ${\displaystyle {}B}$ (or the edge opposite of ${\displaystyle {}C}$). It is often denoted by ${\displaystyle {}c}$, its length is

${\displaystyle {}d(A,B)=\Vert {\overrightarrow {AB}}\Vert \,.}$

The corresponding denominations hold for the other edges. Sometimes we also denote the lengths of the edges by ${\displaystyle {}a,b,c}$. The angle ${\displaystyle {}\angle (A,B,C)}$ of the triangle at the vertex ${\displaystyle {}B}$ is defined as

${\displaystyle \angle ({\overrightarrow {BA}},{\overrightarrow {BC}}),}$

accordingly for the other vertices. The angles are usually denoted by ${\displaystyle {}\alpha ,\beta ,\gamma }$.

We also say that congruent triangles can be transformed into each other by an affine-linear isometry.