You have a bisector of an angle in triangle ABC (eg the bisector of angle A) and it intersects the opposite side BC at point D. We know that then AB+CD=AC+BD is valid.  Does the opposite direction also apply, i.e.  if it is true that AB+CD=AC+BD, then AD is the bisector of angle A?

Yes, the opposite direction also applies. If it is true that \( AB + CD = AC + BD \) in triangle \( ABC \) with \( D \) lying on \( BC \), then \( AD \) is the angle bisector of \( \angle A \).

This result is a consequence of the Angle Bisector Theorem, which states that the angle bisector of an angle in a triangle divides the opposite side into two segments that are proportional to the lengths of the other two sides. Specifically, if \( AD \) is the bisector of \( \angle A \) in triangle \( ABC \), then:

\[ \frac{BD}{DC} = \frac{AB}{AC} \]

Let's prove the converse:

Given:
\[ AB + CD = AC + BD \]

We need to show that \( AD \) is the angle bisector of \( \angle A \).

Rewrite the given equation:
\[ AB + CD = AC + BD \]
\[ AB - BD = AC - CD \]

Let's denote \( BD = x \) and \( DC = y \). Then \( BC = x + y \).

Rewrite the equation again:
\[ AB - BD = AC - DC \]
\[ AB - x = AC - y \]
\[ AB - AC = x - y \]

Thus:
\[ x - y = AB - AC \]

To use the Angle Bisector Theorem:
\[ \frac{BD}{DC} = \frac{x}{y} = \frac{AB}{AC} \]

Since \( AD \) is the bisector of \( \angle A \):
\[ x = \frac{AB \cdot BC}{AB + AC} \]
\[ y = \frac{AC \cdot BC}{AB + AC} \]

Thus:
\[ \frac{BD}{DC} = \frac{AB \cdot BC / (AB + AC)}{AC \cdot BC / (AB + AC)} = \frac{AB}{AC} \]

This verifies that \( \frac{BD}{DC} = \frac{AB}{AC} \), hence \( AD \) must be the angle bisector of \( \angle A \).

So, if \( AB + CD = AC + BD \), then \( AD \) is indeed the bisector of \( \angle A \).

Would you like more details or have any questions?

Here are some related questions:

1. What is the Angle Bisector Theorem?
2. How do you derive the Angle Bisector Theorem?
3. Can the Angle Bisector Theorem be applied to any triangle?
4. How does the Angle Bisector Theorem relate to the Incenter of a triangle?
5. What other properties do angle bisectors in a triangle have?
6. How can you prove the converse of the Angle Bisector Theorem using coordinate geometry?
7. Are there any special cases where the Angle Bisector Theorem does not apply?
8. How does the Angle Bisector Theorem relate to the lengths of the sides of a triangle?

**Tip:** When solving problems involving angle bisectors, remember to look for ways to apply the Angle Bisector Theorem, which often simplifies finding lengths and relationships within the triangle.

Explore the Angle Bisector Theorem in triangle geometry with a detailed explanation and proof. Learn how the theorem applies to triangles, specifically how AB + CD = AC + BD implies AD is the angle bisector of angle A. Suitable for high school students and anyone interested in geometry proofs.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation