Passage: Let $X(a,p)$, $Y(b,q)$ and $Z(c,r)$ be the points such that $a$, $b$ and $c$ are in AP.

Question: If $p$, $q$ and $r$ are in AP, then the points $X$, $Y$ and $Z$ are

Given $a, b, c$ are in AP, let the common difference be $d_1$. Then $b - a = d_1$ and $c - b = d_1$.

Given $p, q, r$ are in AP, let the common difference be $d_2$. Then $q - p = d_2$ and $r - q = d_2$.

The slope of the line segment joining $X(a,p)$ and $Y(b,q)$ is $m_{XY} = \dfrac{q - p}{b - a} = \dfrac{d_2}{d_1}$.

The slope of the line segment joining $Y(b,q)$ and $Z(c,r)$ is $m_{YZ} = \dfrac{r - q}{c - b} = \dfrac{d_2}{d_1}$.

Since $m_{XY} = m_{YZ}$ and the point $Y$ is common to both segments, the points $X$, $Y$, and $Z$ are collinear.

Thus, the points lie on a straight line.

Answer: on a straight line