In triangle ABC, choose points D and E on sides AB and AC so that DE is parallel to the base BC. Then, surprisingly, AD:DB = AE:EC always holds. Even DE:BC follows the same scale factor. Move D with the slider and check that these ratios remain consistent.
Why does this ratio always hold? Because DE is parallel to BC, triangles ADE and ABC are similar (AA similarity: angle A is shared, and corresponding angles are equal for parallel lines, so angle ADE = angle ABC). Corresponding sides of similar triangles always have the same ratio, so AD:AB = AE:AC = DE:BC, and from this we naturally get AD:DB = AE:EC.
This property also works in reverse — if AD:DB = AE:EC, then DE must be parallel to BC. So it can also be used as a tool for checking whether two lines are parallel.