Answer
$(\frac{b}{3},\frac{c}{3})$
Work Step by Step
To find the point of intersection, we must find the equation of at least 2 of the medians. That can be found by calculating the midpoints of each side and then the slope of each median.
1. Equation of median passing through the top corner:
$midpoint=(\frac{-a+a}{2},\frac{0+0}{2})=(0,0)$
$slope=\frac{c-0}{b-0}=\frac{c}{b}$
Equation:
$y-0=\frac{c}{b} (x-0)$
$y=\frac{cx}{b}$
2. Equation of median passing through the bottom-left corner:
$midpoint = (\frac{a+b}{2},\frac{c+0}{2})= (\frac{a+b}{2},\frac{c}{2})$
$slope=\frac{\frac{c}{2}-0}{\frac{a+b}{2}+a}=\frac{\frac{c}{2}}{\frac{3a+b}{2}}=\frac{c}{3a+b}$
Equation:
$y-\frac{c}{2}=\frac{c}{3a+b}(x-\frac{a+b}{2})$
Now we can substitute $y = \frac{cx}{b}$ to solve for x:
$\frac{cx}{b}-\frac{c}{2}=\frac{c}{3a+b}(x-\frac{a+b}{2})$
$\frac{2cx}{2b}-\frac{bc}{2b}=\frac{c}{3a+b} (\frac{2x}{2}-\frac{a+b}{2})$
$\frac{2cx-bc}{2b}=\frac{c}{3a+b} (\frac{2x-a-b}{2})$
$\frac{2x-b}{b}=\frac{2x-a-b}{3a+b}$ [multiply both sides by $\frac{2}{c}$]
$6ax-3ab+2xb-b^2=2xb-ab-b^2$ [cross multiply]
$6ax=2ab$
$x=\frac{b}{3}$
Since we have solved x, it is time to go back and solve y:
$y=\frac{cx}{b}$
$y=\frac{c}{b}\frac{b}{3}$
$y=\frac{c}{3}$
The intersection is $(\frac{b}{3},\frac{c}{3})$
