ABCD is a rectangle. $AB=1$, $AD=\sqrt{3}$. Using $B$ as the center and $BA$ as the radius, draw a circle that intersects $BC$ at $E$. $P$ is a point on arc $AE$. Draw circle $B$'s tangent line through point $P$ that intersects $AD$ at $S$ and $BC$ at $T$. If $ST$ bisects $ABCD$ into two parts of equal area, how long is $ST$?
Triangles $ADB$ and $DBC$ are 30-60-90 triangles, but I don't know how to use the information. I tried assuming that point $P$ is on diagonal $DB$ and solved to get $ST$ is $\frac{2\sqrt{3}}{3}$ but I don't think my teacher will accept that as a valid solution. Please help me out!
You're absolutely right. A line segment bisects a rectangle into two regions of equal area if and only if it passes through the center $O$ of the rectangle. You can show by the Pythagorean theorem that $BO=1$, so it lies on the arc $AE$ and therefore the only possible point $P$ where the tangent will pass through $O$ is $P=O$ itself.
Good job saving yourself a half-ton of ugly algebra!