Law of Cosines

In triangle , the following equations hold:

Proof:
Let be the foot of the perpendicular dropped from vertex to the line . We will prove that the equation holds for three cases: when is acute, right, or obtuse.

Case i. (Acute Angle)

In the right triangle :

Also,

In the right triangle , applying the Pythagorean theorem:

Substitute the values:

Since , we get:

Case ii. (Right Angle)

In the right triangle , , so:

Case iii. (Obtuse Angle)

In the right triangle :

Also,

In the right triangle , applying the Pythagorean theorem:

Since , we get:

From cases i, ii, and iii, we conclude that:

holds for all values of .