Must be a math class .....

Since there are $17$ girls seated in each row there are $17r$ chairs occupied by girls, and since there are $14$ boys seated in each column there are $14c$ chairs occupied by boys. There are a total of $rc$ chairs in the lecture hall, so with $9$ empty chairs we can form the equation

$17r + 14c + 9 = rc \Longrightarrow 14c + 9 = r(c - 17)$

$\Longrightarrow r = \dfrac{14c + 9}{c - 17} = \dfrac{14c - (14)(17) + (14)(17) + 9}{c - 17} = 14 + \dfrac{247}{c - 17}.$

Now $r$ must be an integer, so $c - 17$ must divide $243.$ Since $247 = 13*19$ , we can have $c - 17$ being $1, 13, 19$ or $243.$ This gives us four possible pairs $(c,r)$ , namely

$(18, 251), (30,33), (36,27), (264,15).$

The resulting products are $4518, 990, 972, 3960$ , and so the desired minimum value of $r*c$ is $\boxed{972}.$