Law of Sines

In a triangle , if the radius of its circumcircle is , then:

Note: In triangle , let the angles be denoted by , , , and the lengths of the sides opposite these angles by respectively.

Proof:
Let be the center of the circumcircle of triangle . We can prove the equation by considering three cases where is acute, right, or obtuse.

Case i. (Acute Angle)

Draw the diameter through and the center . Since , we have:

In triangle , since :

Thus, , or .

Case ii: (Right Angle)

When , , so .
Thus, .

Case iii. (Obtuse Angle)

Draw the diameter through and the center . Since , we have:

In triangle , since :

Thus, , or .

From cases i, ii, and iii, we conclude that the equation holds regardless of the size of .