## Morley's Theorem

### Law of Sines

Given any triangle $$[ABC]$$, it is known that its perpendicular bisectors (lines whose points are equidistant from the ends of a side of the triangle) intersect at a point $$O$$, called the circumcenter, which is equidistant from the vertices $$A$$, $$B$$ and $$C$$, that is, such that $$\overline{OA} = \overline{OB} = \overline{OC}$$. It is therefore possible to construct a circle with center at this point, passing through the three vertices of the triangle. The radius of this circle is called circumradius $$(R)$$.

The law of sines tells us that there is a proportionality between the length of each side of the triangle and the sine of the opposite angle, and the constant of proportionality is twice the circumradius. $\frac{\overline{AB}}{\sin (C)}= \frac{\overline{BC}}{\sin (A)} =\frac{ \overline{CA}}{\sin (B)} = 2R$