Video narration
Why does a line parallel to one side of a triangle divide the other two sides in the same ratio? Picture ABC: D lies on AB, E on AC, and DE is parallel to BC. The cut does more than make a smaller triangle: it makes a same-shaped one. That's similar triangles: same shape, so corresponding sides scale together. With DE parallel to BC, AD over AB equals AE over AC. That leads to the side pieces: AD over DB equals AE over EC. The Basic Proportionality Theorem says a parallel cut divides those pieces proportionally. If AD is 3, DB is 2, and AE is 6, then EC is 4: both ratios are 3 to 2. Remember, the line must be parallel to BC; another cut doesn't guarantee this relationship. So why the same ratio? The parallel cut preserves the triangle's shape, so the side pieces stay in proportion.