The Side-Splitter Theorem

Draw a line across a triangle parallel to one of its sides. It cuts the other two sides into pieces that are in the same ratio.

If the line splits one side into 3 and 6, it splits the other into pieces with the same 1:2 ratio.

The converse is also true and is how you prove a line is parallel: if the two sides are divided proportionally, the line must be parallel to the third side.

Watch which pieces you are comparing. The theorem relates the *parts* to each other — top piece to bottom piece on each side. Comparing a part to the whole side works too, as long as you do the same on both sides. Mixing the two is the usual mistake.

A related result: three or more parallel lines cut any two transversals proportionally.

The Side-Splitter (Basic Proportionality or Thales') Theorem states that if a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally: AD/DB = AE/EC.

It follows from AA similarity, since the parallel line creates a smaller triangle with congruent corresponding angles, hence △ADE ∼ △ABC. That similarity gives AD/AB = AE/AC, which rearranges to the part-to-part form. Both forms are valid; the error is mixing part-to-part on one side with part-to-whole on the other.

The converse holds and is the standard tool for proving parallelism from length information alone.

Two extensions appear in problems. Three parallel lines cutting two transversals divide them proportionally, and a triangle's angle bisector divides the opposite side into segments proportional to the adjacent sides — BD/DC = AB/AC — which is proved by constructing an auxiliary parallel line.