1. **Problem statement:** Given points $P(x_1,y_1)$, $Q(x_2,y_2)$, and $R(x,y)$ lie on the circumference of a circle with $PQ$ as the diameter, show the equation of the circle. 2. **Formula and concept:** The circle with diameter $PQ$ has its center at the midpoint of $P$ and $Q$, and radius equal to half the distance $PQ$. 3. **Midpoint of $PQ$:** $$ M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) $$ 4. **Radius of the circle:** $$ r = \frac{1}{2} \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$ 5. **Equation of the circle:** $$ (x - \frac{x_1+x_2}{2})^2 + (y - \frac{y_1+y_2}{2})^2 = r^2 $$ 6. **Using the property of a circle with diameter $PQ$:** Any point $R(x,y)$ on the circle satisfies the right angle property: $$ \angle PRQ = 90^\circ $$ 7. **Using the dot product for perpendicular vectors:** Vectors $\overrightarrow{RP} = (x_1 - x, y_1 - y)$ and $\overrightarrow{RQ} = (x_2 - x, y_2 - y)$ are perpendicular, so: $$ \overrightarrow{RP} \cdot \overrightarrow{RQ} = 0 $$ 8. **Dot product expansion:** $$ (x_1 - x)(x_2 - x) + (y_1 - y)(y_2 - y) = 0 $$ 9. **Rearranging terms:** $$ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 $$ This is the equation of the circle with $PQ$ as diameter. **Final answer:** $$ (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0 $$