The centroid

A median joins a corner to the midpoint of the opposite side. Every triangle has three.

Draw all three: they always pass through one point, the centroid. (Cut a triangle from card: it balances on a pencil tip at the centroid.)

The centroid divides each median 2 : 1, the longer part next to the corner: CM = 2MP. So it sits two-thirds of the way from each corner. A 9 cm median is split into 6 cm and 3 cm.

The proof uses the Midpoint Theorem twice. Let medians CP and BQ meet at M, and let X and Y be the midpoints of BM and CM:

• in ΔABC, PQ ‖ BC and PQ = BC/2;
• in ΔMBC, XY ‖ BC and XY = BC/2;
• so ΔMPQ ≅ ΔMYX (ASA), and MP = MY = YC.

Why all three meet. The same argument for medians AR and BQ gives a point N with BN : NQ = 2 : 1. Only one point divides BQ that way, so N = M.

A second proof: extend AM to S with MS = AM. BSCM is a parallelogram, and its diagonals bisect each other.