Section Formula — Internal Division

The complex number dividing the segment joining z₁ and z₂ internally in the ratio m:n. Identical in structure to the Cartesian section formula, with complex numbers replacing coordinates directly.

The section formula in the complex plane is structurally identical to its Cartesian form — complex numbers replace coordinate pairs, but the derivation is the same.

Setup

Let $P$ divide the segment joining $z_1$ and $z_2$ internally in the ratio $m : n$. This means:

$$\frac{|P - z_1|}{|P - z_2|} = \frac{m}{n}$$

and $P$ lies between $z_1$ and $z_2$ on the segment (same direction from both endpoints).

Derivation

Since $P$ divides $z_1 z_2$ internally in ratio $m:n$, the vector from $z_1$ to $P$ is the fraction $m/(m+n)$ of the total vector from $z_1$ to $z_2$:

$$P - z_1 = \frac{m}{m+n}(z_2 - z_1)$$

Solving for $P$:

$$P = z_1 + \frac{m}{m+n}(z_2 - z_1) = \frac{(m+n)z_1 + m(z_2 - z_1)}{m+n} = \frac{nz_1 + mz_2}{m+n}$$ $$\boxed{P = \frac{mz_2 + nz_1}{m + n}}$$

Verification

Check the ratio:

$$P - z_1 = \frac{mz_2 + nz_1}{m+n} - z_1 = \frac{m(z_2 - z_1)}{m+n}$$ $$P - z_2 = \frac{mz_2 + nz_1}{m+n} - z_2 = \frac{-n(z_2 - z_1)}{m+n}$$ $$\frac{P - z_1}{P - z_2} = \frac{m(z_2 - z_1)}{-n(z_2 - z_1)} = -\frac{m}{n}$$

The negative sign confirms $P$ lies between $z_1$ and $z_2$ (they point in opposite directions from $P$), and the ratio of distances is $m:n$. $\blacksquare$

Special Cases

Midpoint ($m = n$):

$$P = \frac{z_1 + z_2}{2}$$

External division in ratio $m:n$ ($m \neq n$): Replace $n$ with $-n$:

$$P = \frac{mz_2 - nz_1}{m - n}$$

The formula has the same structure — the sign flip accounts for $P$ lying outside the segment rather than between the endpoints.