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

Proof

 * Straight-line-2-points-form-Proof-3.png

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: