Properties of Triangle

Every triangle has six basic elements — its three sides and three angles — and solving a triangle means finding all six once enough of them are known. The Pythagorean theorem handles a right triangle completely, but for an oblique triangle (one with no right angle) two further tools are needed: the Law of Sines and the Law of Cosines. This section develops both, together with the projection formula, the area formula, and the half-angle/Heron's formulas that follow from them.

Notation. For a triangle $ABC$, the angles at the vertices $A,B,C$ are themselves called $A,B,C$, and the side opposite each vertex is named with the corresponding lowercase letter: $a=BC$ (opposite $A$), $b=CA$ (opposite $B$), $c=AB$ (opposite $C$). The symbol $\triangle$ (used alone, without a vertex label) denotes the area of the triangle.

Circumcircle. The circle passing through all three vertices of a triangle is its circumcircle; its centre is the circumcentre and its radius $R$ is the circumradius. This circumradius $R$ turns out to be exactly the constant that appears on the right-hand side of the Law of Sines — that is the reason the circumcircle is introduced here.

Two standing facts about any triangle, used constantly through the rest of this section:

• The three angles always sum to a straight angle: $A+B+C=\pi$ (i.e. $180^\circ$). …