Question 5
(i) $ABCD$ is a square with side $1$. $M$ is the midpoint of $AB$, and $N$ is the midpoint of $BC$. The lines $CM$ and $DN$ meet at $I$. Find the area of the triangle $CIN$.
(ii) The midpoints of the sides $AB$, $BC$, $CD$, $DA$ of the parallelogram $ABCD$ are $M$, $N$, $P$, $Q$ respectively. Each midpoint is joined to the two vertices not on its side. Show that the area outside the resulting 8-pointed star is $2/5$ the area of the parallelogram.
(iii) $ABC$ is a triangle with $CA = CB$ and centroid $G$. Show that the area of $AGB$ is $1/3$ of the area of $ABC$.
(iv) Is (ii) true for all convex quadrilaterals $ABCD$?
(ii) The midpoints of the sides $AB$, $BC$, $CD$, $DA$ of the parallelogram $ABCD$ are $M$, $N$, $P$, $Q$ respectively. Each midpoint is joined to the two vertices not on its side. Show that the area outside the resulting 8-pointed star is $2/5$ the area of the parallelogram.
(iii) $ABC$ is a triangle with $CA = CB$ and centroid $G$. Show that the area of $AGB$ is $1/3$ of the area of $ABC$.
(iv) Is (ii) true for all convex quadrilaterals $ABCD$?