State and prove the interior angle bisector theorem.

The Interior Angle Bisector Theorem is a fundamental concept in geometry that relates to triangles. It states that the angle bisector of an angle in a triangle divides the opposite side into segments that are proportional to the lengths of the other two sides.

Theorem Statement

In triangle ABC, if a point D lies on side BC such that AD is the angle bisector of angle A, then:

• BD/DC = AB/AC

Proof of the Theorem

To prove this theorem, we can use the concept of similar triangles. Here’s a step-by-step breakdown:

1. Draw triangle ABC with angle A being bisected by line AD, meeting side BC at point D.
2. Drop a perpendicular from point A to line BC, and let this intersection be point E.
3. Now, triangles ABE and ACD are formed.
4. Since AD bisects angle A, we have angle BAD = angle CAD.
5. Both triangles share angle A, making triangles ABE and ACD similar by the Angle-Angle (AA) criterion.

From the similarity of triangles, we can write:

• AB/AC = BE/CD

Rearranging gives us:

• BD/DC = AB/AC

This completes the proof of the Interior Angle Bisector Theorem, demonstrating that the segments created by the angle bisector are proportional to the lengths of the other two sides of the triangle.