Equation of Straight Line in Plane/Two-Point Form/Proof 3
Jump to navigation
Jump to search
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