Determine the equivalent resistance between points a and b for the...

Label the junction between the first and second resistor as $c$, and the junction between the second and third resistor as $d$.

A connecting wire joins node $a$ to node $d$, making them equipotential. Thus, node $d$ acts as node $a$.
Another connecting wire joins node $c$ to node $b$, making them equipotential. Thus, node $c$ acts as node $b$.

The first resistor is connected between node $a$ and node $c$ (node $b$).
The second resistor is connected between node $c$ (node $b$) and node $d$ (node $a$).
The third resistor is connected between node $d$ (node $a$) and node $b$.

All three resistors are connected in parallel between terminals $a$ and $b$.

The equivalent resistance $R_{eq}$ is calculated as:
$\dfrac{1}{R_{eq}} = \dfrac{1}{r} + \dfrac{1}{r} + \dfrac{1}{r}$

$\dfrac{1}{R_{eq}} = \dfrac{3}{r}$

$R_{eq} = \dfrac{r}{3}$