Rectangle in triangle

First we put a line of length $a$ on the x-axis, and a line of length $b$ on the y-axis. Connecting the endpoints $(a,0)$ and $(0,b)$ , we get a line with equation $y=b-\frac{b}{a}x$ . These lines form a triangle. Then, we draw a line at $x=c$ , for some $c$ . We see that the width of the triangle is $c$ , while the height is $b-\frac{b}{a}c$ . Thus, the area is $c(b-\frac{b}{a}c)=bc-\frac{b}{a}c^2$ , a parabolic equation. The maximum of a parabola with form $y=Ax^2+Bx+C$ is $(-\frac{B}{2A},C-\frac{B^2}{4A})$ , so the maximum area is $0-\frac{(b)^2}{4\frac{-b}{a}} = -\frac{-ab^2}{4b}=\frac{ab}{4}$ . This corresponds to cutting each length in half and drawing a rectangle to the edge.