the midpoint formula
Where is the exact middle of the line segment joining two points? On a single number line, the midpoint of 2 and 8 is their average, (2 + 8)/2 = 5 — the balance point. The midpoint formula does this on the plane by averaging the two coordinates separately, because the x and the y march independently.
Given the endpoints P_1 = (x_1, y_1) and P_2 = (x_2, y_2), the midpoint M is M = ((x_1 + x_2)/2, (y_1 + y_2)/2). In words: the midpoint's x is the average of the two x's, and its y is the average of the two y's. For example, the midpoint of (1, 2) and (5, 8) is ((1 + 5)/2, (2 + 8)/2) = (3, 5). You can check it sits halfway: it is the same distance from each end and lies on the segment.
This is the section formula's simplest case — dividing the segment in the ratio 1:1 — and it is a workhorse in coordinate proofs: the midpoint of the diagonals of a parallelogram, the center of a circle given a diameter's endpoints, or the apex you place to bisect a base. A common slip is to subtract instead of add; the midpoint uses sums (averages), while the distance formula uses differences — keep the two straight.
A parallelogram has vertices P(1, 1), Q(7, 2), R(8, 6), S(2, 5). The diagonals are PR and QS. Midpoint of PR = ((1+8)/2, (1+6)/2) = (4.5, 3.5); midpoint of QS = ((7+2)/2, (2+6)/2) = (4.5, 4). They differ, so this is not a parallelogram after all — the midpoints would have to coincide.
Diagonals of a parallelogram bisect each other, so their midpoints match — a quick coordinate test.
The midpoint divides the segment in the ratio 1:1; for any other split point use the section formula. And the midpoint always lies on the segment — it is between the endpoints, never outside.