Angle Bisector Theorem
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Introduction & Formulas
Further by combining with Stewart's Theorem it can be shown that
By the Law of Sines on and ,
First, because is an angle bisector, we know that and thus , so the denominators are equal.
Second, we observe that and . Therefore, , so the numerators are equal.
It then follows that
Examples & Problems
- Let ABC be a triangle with angle bisector AD with D on line segment BC. If and , find AB and AC.
Solution: By the angle bisector theorem, or . Plugging this into and solving for AC gives . We can plug this back in to find .
- In triangle ABC, let P be a point on BC and let . Find the value of .
Solution: First, we notice that . Thus, AP is the angle bisector of angle A, making our answer 0.
- Part (b), 1959 IMO Problems/Problem 5.