The Angle Bisector Theorem
by aoum, Mar 16, 2025, 10:06 PM
The Angle Bisector Theorem: A Fundamental Result in Geometry
The Angle Bisector Theorem is a fundamental theorem in Euclidean geometry that describes a relationship between the sides of a triangle and the segments formed by an angle bisector. This theorem is particularly useful in solving geometric problems involving proportionality and triangle ratios.
1. Statement of the Angle Bisector Theorem
In a triangle
, let the angle bisector of 
intersect side 
at a point 
. Then, the Angle Bisector Theorem states:
![\[
\frac{AB}{AC} = \frac{BD}{DC}.
\]](//latex.artofproblemsolving.com/3/b/e/3be878e49e613395d89601202f5cfeb54f3629dd.png)
In other words, the angle bisector divides the opposite side into two segments that are proportional to the lengths of the other two sides.
Additionally, if the angle bisector is internal (dividing the angle inside the triangle), the following relationships hold:
![\[
BD = \frac{AB \cdot BC}{AB + AC}, \quad DC = \frac{AC \cdot BC}{AB + AC}.
\]](//latex.artofproblemsolving.com/7/2/9/729d28b285e0da9dfc9aabacc173a99998aa89b4.png)
2. Proof of the Angle Bisector Theorem
We will prove the theorem using the concept of similar triangles.
Consider
and 
:
1. Since
is the angle bisector, we have:
![\[
\angle BAD = \angle CAD.
\]](//latex.artofproblemsolving.com/8/7/4/8742e07116c14942783641c9580c8ae337caa827.png)
2.
(vertically opposite angles are equal).
By the Angle-Angle (AA) Similarity Criterion, we conclude:
![\[
\triangle ABD \sim \triangle ACD.
\]](//latex.artofproblemsolving.com/d/0/1/d01a21c09febcde7738a9ec7d28235d58df48e54.png)
Since the triangles are similar, corresponding side lengths are proportional:
This completes the proof.
3. Converse of the Angle Bisector Theorem
The converse is also true:
If a point on side of 
satisfies:
![\[
\frac{AB}{AC} = \frac{BD}{DC},
\]](//latex.artofproblemsolving.com/7/2/6/72623ce405ac96b37c9ab8428aabf8cb1ef52c93.png)
then
is the angle bisector of .
Proof: Using the same argument in reverse, we can show that
, implying that bisects the angle.
4. The Exterior Angle Bisector Theorem
If an external angle bisector of intersects side at , the analogous relationship holds:
but this time, the segments
and 
are on opposite sides of the extension.
5. Applications of the Angle Bisector Theorem
The Angle Bisector Theorem is useful in several geometric contexts:
Example Problem 1:
In, the angle bisector of meets at . If 
, 
, and 
, find the lengths of and .
Solution:
By the Angle Bisector Theorem:
![\[
\frac{BD}{DC} = \frac{AB}{AC} = \frac{8}{6} = \frac{4}{3}.
\]](//latex.artofproblemsolving.com/c/8/d/c8d6c998a33a99a40c482cc21d5f2f2efab6bf27.png)
Let
and 
. Since 
:
![\[
4x + 3x = 14 \implies 7x = 14 \implies x = 2,
\]](//latex.artofproblemsolving.com/1/9/6/1968c045c78a41edf5a09cb8061f3a2cf9301884.png)
So,
![\[
BD = 4(2) = 8, \quad DC = 3(2) = 6.
\]](//latex.artofproblemsolving.com/1/d/f/1df521beefe3c57b5cd5217333199c72369d224c.png)
Answer:
, 
.
Example Problem 2:
Prove that the internal and external angle bisectors of an angle of a triangle are perpendicular.
Solution:
Let be the internal angle bisector and 
be the external angle bisector of . By construction, these two lines divide the angle into complementary parts:
![\[
\angle BAD + \angle BAE = 90^\circ.
\]](//latex.artofproblemsolving.com/5/e/a/5ea1380bbdcf277b58630ee90cf8211b9e8d3bb9.png)
Thus, they are perpendicular.
6. Generalized Angle Bisector Theorem
For any cevian (a line segment from a vertex to the opposite side) dividing the opposite side in a ratio
, the following holds:
![\[
\frac{AB}{AC} = \frac{m}{n}.
\]](//latex.artofproblemsolving.com/2/6/1/2619d026e0b6e178311cc6b5ad34c8c4d9fb1e84.png)
The angle bisector is a special case where
and 
.
7. Conclusion
The Angle Bisector Theorem is a powerful tool in geometry, linking angle division with side ratios. Its simplicity and versatility make it indispensable in solving advanced geometry problems.
References
The Angle Bisector Theorem is a fundamental theorem in Euclidean geometry that describes a relationship between the sides of a triangle and the segments formed by an angle bisector. This theorem is particularly useful in solving geometric problems involving proportionality and triangle ratios.
1. Statement of the Angle Bisector Theorem
In a triangle
, let the angle bisector of 
intersect side 
at a point 
. Then, the Angle Bisector Theorem states:![\[
\frac{AB}{AC} = \frac{BD}{DC}.
\]](http://latex.artofproblemsolving.com/3/b/e/3be878e49e613395d89601202f5cfeb54f3629dd.png)
In other words, the angle bisector divides the opposite side into two segments that are proportional to the lengths of the other two sides.
Additionally, if the angle bisector is internal (dividing the angle inside the triangle), the following relationships hold:
![\[
BD = \frac{AB \cdot BC}{AB + AC}, \quad DC = \frac{AC \cdot BC}{AB + AC}.
\]](http://latex.artofproblemsolving.com/7/2/9/729d28b285e0da9dfc9aabacc173a99998aa89b4.png)
2. Proof of the Angle Bisector Theorem
We will prove the theorem using the concept of similar triangles.
Consider
and 
:1. Since
is the angle bisector, we have:![\[
\angle BAD = \angle CAD.
\]](http://latex.artofproblemsolving.com/8/7/4/8742e07116c14942783641c9580c8ae337caa827.png)
2.
(vertically opposite angles are equal).By the Angle-Angle (AA) Similarity Criterion, we conclude:
![\[
\triangle ABD \sim \triangle ACD.
\]](http://latex.artofproblemsolving.com/d/0/1/d01a21c09febcde7738a9ec7d28235d58df48e54.png)
Since the triangles are similar, corresponding side lengths are proportional:
This completes the proof.
3. Converse of the Angle Bisector Theorem
The converse is also true:
If a point
on side of 
satisfies:![\[
\frac{AB}{AC} = \frac{BD}{DC},
\]](http://latex.artofproblemsolving.com/7/2/6/72623ce405ac96b37c9ab8428aabf8cb1ef52c93.png)
then
is the angle bisector of .Proof: Using the same argument in reverse, we can show that
, implying that bisects the angle.4. The Exterior Angle Bisector Theorem
If an external angle bisector of
intersects side at , the analogous relationship holds:but this time, the segments
and 
are on opposite sides of the extension.5. Applications of the Angle Bisector Theorem
The Angle Bisector Theorem is useful in several geometric contexts:
- Finding Unknown Lengths: If you know the lengths of two sides and one segment, you can calculate the other segment using proportionality.
- Proving Similarity and Congruence: It can be used to establish the similarity of triangles in more complex geometric constructions.
- Solving Geometry Olympiad Problems: Many competition problems use the angle bisector theorem to derive key relationships.
Example Problem 1:
In
, the angle bisector of meets at . If 
, 
, and 
, find the lengths of and .Solution:
By the Angle Bisector Theorem:
![\[
\frac{BD}{DC} = \frac{AB}{AC} = \frac{8}{6} = \frac{4}{3}.
\]](http://latex.artofproblemsolving.com/c/8/d/c8d6c998a33a99a40c482cc21d5f2f2efab6bf27.png)
Let
and 
. Since 
:![\[
4x + 3x = 14 \implies 7x = 14 \implies x = 2,
\]](http://latex.artofproblemsolving.com/1/9/6/1968c045c78a41edf5a09cb8061f3a2cf9301884.png)
So,
![\[
BD = 4(2) = 8, \quad DC = 3(2) = 6.
\]](http://latex.artofproblemsolving.com/1/d/f/1df521beefe3c57b5cd5217333199c72369d224c.png)
Answer:
, 
.Example Problem 2:
Prove that the internal and external angle bisectors of an angle of a triangle are perpendicular.
Solution:
Let
be the internal angle bisector and 
be the external angle bisector of . By construction, these two lines divide the angle into complementary parts:![\[
\angle BAD + \angle BAE = 90^\circ.
\]](http://latex.artofproblemsolving.com/5/e/a/5ea1380bbdcf277b58630ee90cf8211b9e8d3bb9.png)
Thus, they are perpendicular.
6. Generalized Angle Bisector Theorem
For any cevian (a line segment from a vertex to the opposite side) dividing the opposite side in a ratio
, the following holds:![\[
\frac{AB}{AC} = \frac{m}{n}.
\]](http://latex.artofproblemsolving.com/2/6/1/2619d026e0b6e178311cc6b5ad34c8c4d9fb1e84.png)
The angle bisector is a special case where
and 
.7. Conclusion
The Angle Bisector Theorem is a powerful tool in geometry, linking angle division with side ratios. Its simplicity and versatility make it indispensable in solving advanced geometry problems.
References
- Wikipedia: Angle Bisector Theorem
- Coxeter, H. S. M. – Introduction to Geometry.
- AoPS Wiki: Angle Bisector Theorem
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