Angle Bisector Theorem Proof

Michael Sow

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28-11-2024

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The Angle Bisector Theorem is a fundamental concept in geometry, stating a crucial relationship between an angle bisector of a triangle and the lengths of the sides it intersects. Understanding its proof is key to mastering many geometric problems. Let's delve into a clear and concise demonstration.

Understanding the Theorem

Before proving it, let's clearly state the theorem: The Angle Bisector Theorem states that an angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle.

More formally, consider a triangle ∆ABC. If AD is the angle bisector of ∠BAC, then AB/AC = BD/DC.

Proof using Similar Triangles

This proof elegantly uses the concept of similar triangles.

1. Constructing Auxiliary Lines: Draw a line parallel to AD through point C, intersecting the extension of line AB at point E.

2. Identifying Similar Triangles: Observe that because AD || CE, we have:

   • ∠BAD = ∠AEC (corresponding angles)
   • ∠CAD = ∠ACE (alternate interior angles)

Since AD bisects ∠BAC, ∠BAD = ∠CAD. Therefore, ∠AEC = ∠ACE, making ∆ACE an isosceles triangle, meaning AC = AE.

3. Applying Similar Triangle Properties: Now, consider triangles ∆ABD and ∆EBC. We can see that:
   • ∠BAD = ∠BEC (corresponding angles)
   • ∠ABD = ∠EBC (common angle)

Therefore, ∆ABD ~ ∆EBC (AA similarity).

4. Establishing the Proportion: From the similarity of ∆ABD and ∆EBC, we get the following proportion of corresponding sides:

AB/BC = BD/DC

This directly proves the Angle Bisector Theorem.

Alternative Proof using Area

Another method involves using the areas of triangles.

1. Area Relationship: The area of a triangle can be expressed as (1/2) * base * height. If we consider the areas of ∆ABD and ∆ACD, sharing the same height from vertex A, we have:

Area(∆ABD) / Area(∆ACD) = (1/2 * AB * h) / (1/2 * AC * h) = AB/AC

2. Area using different bases: Alternatively, considering the same height from vertex A, we can write:

Area(∆ABD) / Area(∆ACD) = (1/2 * BD * h) / (1/2 * DC * h) = BD/DC

3. Combining the relationships: Since both expressions represent the ratio of areas of ∆ABD and ∆ACD, we can equate them:

AB/AC = BD/DC

Thus, we have again demonstrated the Angle Bisector Theorem.

Conclusion

The Angle Bisector Theorem, a cornerstone of geometry, holds significant importance in solving various problems related to triangles. Both proofs presented – one leveraging similar triangles and the other employing area relationships – highlight the versatility and elegance of geometrical reasoning. Mastering this theorem enhances one's problem-solving skills within the field of geometry and beyond.