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 an angle bisector in a triangle divides the opposite side into segments that are proportional to the adjacent sides. Specifically, if the angle bisector of angle A in triangle ABC intersects the side BC at point D, then BD/DC = AB/AC.
To prove this theorem, we can use the Law of Sines. Let a, b, and c be the side lengths opposite to angles A, B, and C, respectively. Then we have:
BD/DC = [ABC]/[ACD] (where [ABC] and [ACD] represent the areas of those triangles)
= (1/2) AB AC sin A / (1/2) AC AD sin DAC (using the area formula)
= AB/AD * sin A/sin DAC
Similarly, we can find that AC/AD = BC/BD * sin C/sin CBD
Multiplying these two equations together, we get:
BD/DC * AC/AD = AB/AD * BC/BD * sin A/sin DAC * sin C/sin CBD
Since angles DAC and CBD are equal (they are opposite to the congruent angles ADB and ADC), we can simplify the right-hand side of the equation:
BD/DC * AC/AD = AB/AD * BC/BD
Which is exactly what we wanted to prove.
This theorem is useful in a variety of geometrical problems involving triangles. For example, given two sides and the angle between them, the Angle Bisector Theorem can be used to find the length of the third side. It can also be used to prove properties of triangles, such as the Incenter Excenter Lemma.
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