**General triangle.**
In general form of the triangle
,
the sides are denoted by
,
, and
,
and the angles are denoted by
,
, and
.

**Inscribed angle theorem.**
is called an *inscribed angle*,
and
is called a *central angle*.
Then they have the following relationship:

**Law of sines.**
Let is the radius of the circumscribed circle
of
,
and let be the diameter of the circle.

Since by the inscribed angle theorem, we have ; thus, we obtain

**Law of cosines.**
Let be perpendicular to .
The square length
can be expressed
in terms of the sides and the angle
via the two different right triangles
and
.

(2-1) | ||

(2-2) |

Comparing the rightmost expressions in (2-1)-(2-2),
we obtain
.
In general we have

**Heron's formula.**
Let be the area of
,
and let
.
By using
via the law of cosines,
we obtain

Department of Mathematics