Two-point Form

The figure shows a straight line drawn in the first quadrant of the $xy$-plane. Three distinct points are marked on this line: $P_1(x_1, y_1)$, $P(x, y)$, and $P_2(x_2, y_2)$. The point $P_1$ is the leftmost of the three, $P_2$ is the rightmost, and $P$ lies somewhere between them. The axes are the standard $x$-axis (horizontal) and $y$-axis (vertical). No grid, no shading, no extra curves — just a single line with three labelled points on it.

The physical idea is simple: if you know two points on a line, you can find the equation of that line. The figure makes this concrete by showing a general point $P$ sliding along the line between the two fixed points $P_1$ and $P_2$. Because all three points are collinear, the slope between $P_1$ and $P$ must equal the slope between $P_1$ and $P_2$. That single equality is the entire geometric content of the diagram.

From that equality, the textbook derives the two-point form. The slope of $P_1P$ is $\frac{y - y_1}{x - x_1}$, and the slope of $P_1P_2$ is $\frac{y_2 - y_1}{x_2 - x_1}$. Setting them equal gives:

Rearranging this into the standard two-point form:

Here, $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two given points, and $(x, y)$ is any point on the line. The fraction $\frac{y_2 - y_1}{x_2 - x_1}$ is the slope $m$ of the line. The formula works for any two distinct points — it does not matter which one you call $P_1$ and which $P_2$, as long as you are consistent.