Question: Proof bpt in video
1. The problem is to prove the Basic Proportionality Theorem (BPT), also known as Thales' theorem.
2. The theorem states: If a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally.
3. Consider triangle $ABC$ with a line $DE$ parallel to side $BC$, intersecting $AB$ at $D$ and $AC$ at $E$.
4. We need to prove that $$\frac{AD}{DB} = \frac{AE}{EC}$$.
5. Since $DE \parallel BC$, angles $ADE$ and $ABC$ are equal (corresponding angles).
6. Similarly, angles $AED$ and $ACB$ are equal.
7. Triangles $ADE$ and $ABC$ are similar by AA similarity criterion.
8. From similarity, corresponding sides are proportional:
$$\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}$$
9. Express $AB$ as $AD + DB$ and $AC$ as $AE + EC$.
10. Using the proportionality:
$$\frac{AD}{AD + DB} = \frac{AE}{AE + EC}$$
11. Cross-multiplying:
$$AD \times (AE + EC) = AE \times (AD + DB)$$
12. Expanding:
$$AD \times AE + AD \times EC = AE \times AD + AE \times DB$$
13. Cancel $AD \times AE$ from both sides:
$$AD \times EC = AE \times DB$$
14. Rearranging:
$$\frac{AD}{DB} = \frac{AE}{EC}$$
15. This completes the proof of the Basic Proportionality Theorem.
The theorem shows that a line parallel to one side of a triangle divides the other two sides proportionally, which is a fundamental property in geometry.