Does any line passing through the centroid of a polygon divides its area in half?

Question

I know the answer is no for any triangle. But is it no for all polygons? Is the answer different for convex and concave polygons?

Answer 1

Let some line passing through centroid divide polygon in half. Then if we rotate the line by some angle (small or large), it should continue to divide in half. But we can easily show that area gained and lost by one side may or may not be equal.

This can be done by rotating line by small angle $d\theta$ and noting the change in area $\frac{(r_1^2-r_2^2) d\theta}{2}$ where $r_1, r_2$ are distances of centroid from perimeter along the line. In general this proposition is false as distance of centriod is not same from all points that is $r_1 \neq r_2$ generally.

Answer 2

Not necessarily.

For example the line through the centroid of an equilateral triangle and parallel to the base, divides the area into two parts with proportion of 4 to 5.

That is the area of the top part is 4/5 of the bottom area.