Equation of Straight Line in Plane/Two-Point Form/Proof 3

Theorem

Let $P_1 := \tuple {x_1, y_1}$ and $P_2 := \tuple {x_2, y_2}$ be points in a cartesian plane.

Let $\LL$ be the straight line passing through $P_1$ and $P_2$.

Then $\LL$ can be described by the equation:

$\dfrac {y - y_1} {x - x_1} = \dfrac {y_2 - y_1} {x_2 - x_1}$

or:

$\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$

Proof

Let $P = \tuple {x, y}$ be an arbitrary point on the straight line through $P_1 = \tuple {x_1, y_1}$ and $P_2 = \tuple {x_2, y_2}$.

Construct the straight line $P_1 H K$ perpendicular to the $x$-axis.

We have that $\triangle P_1 H P_2$ and $\triangle P_1 K P$ are similar.

Hence:

\(\ds \dfrac {P_1 K} {P_1 H}\)

\(=\)

\(\ds \dfrac {K P} {H P_2}\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac {x - x_1} {x_2 - x_1}\)

\(=\)

\(\ds \dfrac {y - y_1} {y_2 - y_1}\)

$\blacksquare$

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $4$. Special forms of the equation of a straight line: $(3)$ Line through two points