# Dividing a Triangle into Two Parts of Equal Area with Constraints

Given a triangle $ABC$ and a line labelled 'n' that passes through the triangle but its not parallel to any of the sides, how do we construct a line parallel to 'n' that will divide $A$$B$$C$ into two parts of equal area.

-
You should check this nice answer: math.stackexchange.com/q/235626/4058 –  Javier Álvarez Nov 16 '12 at 12:05

Up to vertex relabelling, we can assume that there is a point $P$ on the $BC$-side such that $AP$ is parallel to $n$ and $BP\leq PC$. The ratio between the areas of $APC$ and $APB$ is equal to $\frac{PC}{PB}$, so it is $\geq 1$. Let $t$ be the ratio between the area of $CAP$ and half the area of $ABC$: $t\geq 1$ holds. If $Q$ lies between $P$ and $C$, the parallel to $n$ through $Q$ cuts $AC$ in $R$ and $\frac{CP}{CQ}=\sqrt{t}$, the area of $CQR$ is half the area of $ABC$.
In summary, if $M$ is the midpoint of $BC$ and $Q$ lies between $P$ and $M$ such that $CQ^2=CM\cdot CP$, the line through $Q$ parallel to $AP$ splits $ABC$ in a triangle and a convex quadrilateral having equal areas.
That won't work unless the line $n$ happens already to be parallel to a bisector of the triangle through one of its vertices, which typically is not the case. –  coffeemath Nov 16 '12 at 5:45