Difference between revisions of "Angle Bisector Theorem"
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The '''Angle Bisector Theorem''' states that given [[triangle]] <math>\triangle ABC</math> and [[angle bisector]] AD, where D is on side BC, then <math> \frac cm = \frac bn </math>. Likewise, the converse of this theorem holds as well. | The '''Angle Bisector Theorem''' states that given [[triangle]] <math>\triangle ABC</math> and [[angle bisector]] AD, where D is on side BC, then <math> \frac cm = \frac bn </math>. Likewise, the converse of this theorem holds as well. | ||
Revision as of 17:22, 14 July 2017
This is an AoPSWiki Word of the Week for June 6-12 |
Contents
Introduction
The Angle Bisector Theorem states that given triangle and angle bisector AD, where D is on side BC, then
. Likewise, the converse of this theorem holds as well.
Introduction
The Angle Bisector Theorem states that given triangle and angle bisector AD, where D is on side BC, then
. Likewise, the converse of this theorem holds as well.
Examples
- 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.