The Midpoint Theorem

Cut a triangle along the lines joining the midpoints of its sides. The four small triangles are congruent, and each is a half-size copy of the original.

The Midpoint Theorem: if P and Q are the midpoints of AB and AC, then PQ ‖ BC and PQ = BC/2. In numbers: BC = 10 gives PQ = 5; BC = 1000 gives PQ = 500.

The proof:

• draw the line through C parallel to BA, meeting line PQ at R;
• ΔAPQ ≅ ΔCRQ (AAS), so PQ = QR and CR = AP = BP;
• CR = BP and CR ‖ BP, so BCRP is a parallelogram;
• so PQ ‖ BC, and PQ = PR/2 = BC/2.

Where did that extra line come from? It can look like magic. Such helper lines (auxiliary constructions) come with experience: try drawing a parallel, a midpoint or a diagonal, and see what appears.

All three midpoints: every small side is half a big side, so the four small triangles are congruent by SSS.