The Basic Proportionality Theorem (BPT), also known as Thales' Theorem, states that:

"If a line is drawn parallel to one side of a triangle and it intersects the other two sides, then it divides those sides in the same proportion."

Given

In a triangle $ABC$, let $DE$ be a line drawn parallel to side $BC$, intersecting $AB$ at $D$ and $AC$ at $E$.

To Prove

$\frac{AD}{DB} = \frac{AE}{EC}$

Proof

1. Draw Triangle and Setup Ratios: Consider $\triangle ABC$ with $DE \parallel BC$. We need to prove that $\frac{AD}{DB} = \frac{AE}{EC}$.

2. Use of Areas: Let's consider the areas of triangles $\triangle ADE$ and $\triangle ABC$.

   Since $DE \parallel BC$, we have two triangles $\triangle ADE$ and $\triangle ABC$ that are similar (by the AA criterion of similarity in triangles).

3. Triangles $\triangle ADE$ and $\triangle ABC$ Are Similar: By the property of similar triangles, corresponding sides are proportional: $\frac{AD}{AB} = \frac{AE}{AC}$

4. Relation Between Segments: Since $DE \parallel BC$, we can express: $\frac{AD}{DB} = \frac{AE}{EC}$ due to the proportionality derived from the similarity.

5. Conclusion: Thus, the Basic Proportionality Theorem is proven, as we have shown that a line drawn parallel to one side of a triangle divides the other two sides in equal proportion.

Would you like further details on any step, or have any specific questions?

Related Questions

1. What are the applications of the Basic Proportionality Theorem in geometry?
2. How can BPT be used to find unknown side lengths in a triangle?
3. Can BPT be applied in quadrilaterals or other polygons?
4. How does BPT relate to the concept of similar triangles?
5. What are some real-life applications where BPT might be useful?

Tip: When working with parallel lines in triangles, consider using BPT to simplify complex ratios.