Why the three medians always meet at one point

2026-06-05

Pick a corner of a triangle. Run a line from it to the exact middle of the side facing it. That line is a median. There are three corners, so there are three medians. Nothing forces three separate lines to share a point - draw three random chords and they make a little triangle of near-misses. But the medians don't miss. They cross at one point, every time.

Drag any vertex anywhere. Make the triangle thin, fat, lopsided, obtuse - it doesn't matter. The three coloured lines refuse to form a gap. The readout below the diagram measures how far apart the three pairwise crossings are; it stays pinned at zero. That stubborn meeting point has a name: the centroid.

One point answers all three

Here is the whole secret, and it fits on one line. Call the three corners A, B, and C, and treat them as positions - pairs of coordinates you can add and average. Consider the single point that is their average:

G = (A + B + C) / 3

Now look at the median from A. It ends at the midpoint of the opposite side, which is (B + C) / 2. Walk two-thirds of the way from A toward that midpoint and you land at

A + ⅔ · ( (B + C)/2 − A ) = (A + B + C) / 3 = G.

The same arithmetic works starting from B, and from C - the formula is symmetric, it doesn't care which corner you call first. So G lies on all three medians at once. Three lines can't avoid a point that every one of them is obliged to pass through. The concurrency isn't a coincidence; it is the symmetry of an average.

The 2 : 1 fall

That "two-thirds of the way" wasn't a fudge - it's exact, and it is the second surprise hiding in the same diagram. On every median, the stretch from the vertex down to G is the bold part; the stretch from G on to the midpoint is the faint dashed part. The bold part is always twice the dashed one. The readout holds at 2.00 : 1 however you drag. The centroid is two-thirds of the way down from every corner and one-third up from every edge, all at the same time.

It is the balance point

An average of positions is a centre of mass. Cut the triangle out of card and it will balance on a pin placed exactly at G - that is what averaging the corners means physically. It is also why a three-legged stool is stable and why the centroid is where engineers hang a triangular plate. The same point that makes the three medians meet is the point the shape would balance on. Two facts, one dot.

The medians are one of four famous triples that each meet at a single point in any triangle - the others belong to the angle bisectors, the perpendicular side-bisectors, and the altitudes. Three of those special points even sit on one straight line, Euler's line. The triangle is far busier on the inside than it lets on.