the section formula

The midpoint splits a segment evenly, but sometimes you want a point that splits it unevenly — say two-thirds of the way along, or in the ratio 2 to 3. The section formula finds the coordinates of that division point. Think of walking from one endpoint toward the other and stopping at a chosen fraction of the journey.

Suppose the point P divides the segment from A = (x_1, y_1) to B = (x_2, y_2) internally in the ratio m:n (meaning AP : PB = m : n). Then P = ((m x_2 + n x_1)/(m + n), (m y_2 + n y_1)/(m + n)). It is a weighted average: notice the weights cross over — the m (which measures the piece nearest A) multiplies the far endpoint x_2. A quick check: if m = n = 1 the formula collapses to ((x_1 + x_2)/2, (y_1 + y_2)/2), the midpoint, exactly as it should. For instance, the point dividing (0, 0) to (10, 5) in ratio 2:3 is ((2*10 + 3*0)/5, (2*5 + 3*0)/5) = (4, 2).

This is the everyday tool for trisection points, centroids (a triangle's medians meet two-thirds of the way from each vertex, the ratio 2:1), and any point at a stated proportion along a segment. There is also an external-division version, where P lies on the line beyond the segment; then one of the ratio numbers is taken negative, for example m:(-n). Getting the cross-over of weights right — m with the far point, n with the near — is the only thing to watch.