Varignon's Theorem: Properties and Proof

• Varignon's Theorem states that the midpoints of the sides of any quadrilateral form a parallelogram.
• This holds true even for non-convex quadrilaterals.
• The parallelogram formed by connecting the midpoints is called the Varignon parallelogram.

Definition and Explanation

• Consider any quadrilateral (a four-sided figure) in a plane.
• Locate the midpoint of each of the four sides of the quadrilateral.
• Connect these midpoints consecutively, forming a new quadrilateral.
• Varignon's Theorem asserts that this new quadrilateral is always a parallelogram, regardless of the shape of the original quadrilateral.
• A parallelogram is a quadrilateral with opposite sides parallel and equal in length.
• The area of the Varignon parallelogram is exactly half the area of the original quadrilateral.
• The perimeter of the Varignon parallelogram is equal to the sum of the diagonals of the original quadrilateral.

Properties of the Parallelogram

• Opposite sides of a parallelogram are parallel and equal in length.
• Opposite angles of a parallelogram are equal.
• Consecutive angles of a parallelogram are supplementary (add up to 180 degrees).
• The diagonals of a parallelogram bisect each other (they intersect at their midpoints).
• Each diagonal of a parallelogram divides it into two congruent triangles.
• The area of a parallelogram is given by the formula: Area = base * height.
• If the original quadrilateral is a rectangle, the Varignon parallelogram is a rhombus.
• If the original quadrilateral is a rhombus, the Varignon parallelogram is a rectangle.
• If the original quadrilateral is a square, the Varignon parallelogram is a square.
• If the original quadrilateral is a parallelogram, the Varignon parallelogram is also a parallelogram.

Proof of the Theorem

• Let ABCD be any quadrilateral.
• Let P, Q, R, and S be the midpoints of sides AB, BC, CD, and DA, respectively.
• Consider triangle ABC and that P and Q are midpoints of AB and BC.
• Segment PQ is parallel to AC and PQ = (1/2)AC by the midpoint theorem, which is also known as the basic proportionality theorem.
• Now consider triangle ADC and that R and S are midpoints of CD and DA.
• Segment SR is parallel to AC and SR = (1/2)AC by the midpoint theorem.
• PQ || AC and SR || AC, therefore PQ || SR.
• PQ = (1/2)AC and SR = (1/2)AC, therefore PQ = SR.
• In quadrilateral PQRS, one pair of opposite sides (PQ and SR) are parallel and equal in length.
• Therefore PQRS is a parallelogram.
• The quadrilateral formed by joining the midpoints of the sides of any quadrilateral is a parallelogram.