Section formula — internal and external division

WHAT are we finding?

HOW — deriving it from scratch

Let $A=(x_1,y_1)$, $B=(x_2,y_2)$, and $P=(x,y)$ divide $AB$ internally in ratio $m:n$.

Why similar triangles? Drop verticals from $A$, $P$, $B$ to the $x$-axis. The horizontal gaps grow in the same proportion as the points move along the line, because the line has a constant slope. So the ratio $AP:PB$ shows up identically in the $x$-shadows and $y$-shadows.

Project onto the $x$-axis. Moving from $A$ to $P$ covers $x-x_1$; from $P$ to $B$ covers $x_2-x$. These are in ratio $m:n$: $\frac{x-x_1}{x_2-x} = \frac{m}{n}.$ Why? Because a straight line splits horizontal distance in exactly the same ratio as it splits the segment length.

Cross-multiply: $n(x-x_1)=m(x_2-x)\;\Rightarrow\; nx-nx_1 = mx_2-mx.$ Collect $x$: $mx+nx = mx_2+nx_1 \;\Rightarrow\; x=\frac{mx_2+nx_1}{m+n}.$ Identically for $y$: $y=\frac{my_2+ny_1}{m+n}.$

External case. Now $P$ is outside, so going $A\to P$ and $P\to B$ point oppositely. One of the projected lengths becomes negative. Replace $n$ by $-n$: $P=\left(\frac{mx_2-nx_1}{m-n},\ \frac{my_2-ny_1}{m-n}\right).$

Midpoint = internal division with $m=n=1$: $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$

Worked examples

Common mistakes

Connections

• Midpoint formula — special case $m=n=1$.
• Centroid of a triangle — divides each median $2:1$ from vertex.
• Similar triangles — the geometric engine of the derivation.
• Distance formula — used to verify ratios independently.
• Slope of a line — constant slope is why the shadow ratios match.
• Collinearity of three points — three points collinear iff one divides the other two in some ratio.

Concept Map

Hinglish (regional understanding)