Varignon's Theorem, a Parallelogram Inside Every Quadrilateral

Definition Varignon’s theorem is a statement about a quadrilateral $ABCD$ with sides $AB$, $BC$, $CD$ and $DA$ and diagonals $AC$ and $BD$. Let $E$, $F$, $G$ and $H$ be the midpoints of $AB$, $BC$, $CD$ and $DA$. The theorem says that the sides $EF$ and $HG$ of the quadrilateral $EFGH$ are both parallel to the diagonal $AC$ and half as long, and that the sides $FG$ and $EH$ are both parallel to the diagonal $BD$ and half as long. So $EFGH$ is a parallelogram, except that it collapses onto a single line when the diagonals $AC$ and $BD$ are parallel, which can happen only when two sides of $ABCD$ cross. When the sides of $ABCD$ do not cross, whether it is convex or not, $EFGH$ is a true parallelogram with nonzero area. When $ABCD$ is convex, the area of $EFGH$ is exactly half the area of $ABCD$. The theorem is named after the French mathematician Pierre Varignon, whose proof was published in 1731, after his death.

Definition A midsegment of a triangle is the segment that joins the midpoints of two of its sides. The triangle midsegment theorem says that in every triangle such a midsegment is parallel to the third side and half as long. One proof shrinks the triangle toward a vertex. Shrinking the plane toward a point $P$ by the factor $\tfrac12$ moves every point $X$ to the midpoint of the segment $PX$. The move from the new position of $X$ to the new position of another point $Y$ is then exactly half the move from $X$ to $Y$, so every segment goes to a parallel segment half as long. Shrinking triangle $ABC$ toward $B$ in this way moves $A$ to the midpoint of $BA$ and $C$ to the midpoint of $BC$, so it sends the side $AC$ onto the midsegment joining those two midpoints, which is therefore parallel to $AC$ and half as long.

Definition The Varignon parallelogram of a quadrilateral $ABCD$ is the parallelogram $EFGH$ whose vertices are the midpoints of $AB$, $BC$, $CD$ and $DA$, in that order. Its sides are parallel to the diagonals $AC$ and $BD$ and half as long, so its perimeter equals $AC + BD$, and its shape depends only on the two diagonals. It is a rhombus exactly when the diagonals have equal length, a rectangle exactly when the diagonals are perpendicular, and a square when both are true. When $ABCD$ is convex, the Varignon parallelogram has half its area. To see this, cut off the four corner triangles $EBF$, $GDH$, $HAE$ and $FCG$ that lie between the parallelogram and the sides of $ABCD$. The triangle $EBF$ is triangle $ABC$ shrunk toward $B$ by the factor $\tfrac12$, which halves its base and its height, so it has a quarter of the area of $ABC$, and in the same way $GDH$ has a quarter of the area of $ACD$. Together these two corners have a quarter of the area of $ABCD$, and the same argument with the diagonal $BD$ shows that $HAE$ and $FCG$ have another quarter. The four corners take up half the area, and the parallelogram is the other half. The half-area rule also holds when one angle of $ABCD$ is greater than $180^\circ$, but then the corner triangle at that vertex lies outside $ABCD$ and is added rather than removed.

Definition A quadrilateral $ABCD$ is made of four points $A$, $B$, $C$ and $D$, its vertices, no three of them on one line, joined in this order by the four sides $AB$, $BC$, $CD$ and $DA$. The segments $AC$ and $BD$ are its diagonals. A quadrilateral is simple when its sides meet only at the shared vertex of two neighboring sides. A simple quadrilateral is convex when each of its angles is less than $180^\circ$, which happens exactly when its two diagonals cross each other, and otherwise one of its angles is greater than $180^\circ$ and it is called concave. When two sides cross, the figure is a crossed quadrilateral, shaped like a bow tie, and it is usually not counted as a polygon. A parallelogram is a quadrilateral whose opposite sides are parallel, and its opposite sides are then also equal in length.