Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle
The Angle Bisector Theorem states that in any triangle, the angle bisector of a vertex divides the opposite side in a ratio proportional to the lengths of the adjacent sides.
More specifically, if we have a triangle ABC, and let D be a point on the segment BC such that AD is the angle bisector of angle A, then we have:
BD/DC = AB/AC
In other words, the ratio of the length of segments BD and DC is equal to the ratio of the length of sides AB and AC.
To see why this is true, we can use the Law of Sines, which states that for any triangle ABC:
a/sin(A) = b/sin(B) = c/sin(C)
If we apply the Law of Sines to triangle ABD and triangle ACD, we get:
AB/sin(BAD) = AD/sin(ABD) … (1)
AC/sin(CAD) = AD/sin(ACD) … (2)
But since angle BAD and angle CAD are supplementary, we know that sin(BAD) = sin(CAD), and by combining equations (1) and (2), we get:
AB/AC = sin(ABD)/sin(ACD) = BD/DC
Thus, we have proven the Angle Bisector Theorem.
Some useful applications of the Angle Bisector Theorem include being able to find the length of sides and/or angles of a triangle given certain information about segment ratios, as well as being able to prove other theorems about triangles.
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