Doubling a median and Varignon's theorem
Extend a median to get a parallelogram and prove a bound for the median; join the midpoints of the sides of any quadrilateral and see what shape appears.
- Russia: grade 8
Guide
In the Russian programme this belongs to Grade 8 geometry: parallelograms, the midline of a triangle, extra constructions. Varignon's theorem and doubling the median appear in harder problems and clubs.
Doubling the median: extend the median AM beyond M by MD = AM. The diagonals of ABDC bisect each other at M, so ABDC is a parallelogram and BD = AC. The triangle inequality in ABD gives AD < AB + BD, so 2·AM < AB + AC.
Varignon's theorem: the midpoints of the sides of ANY quadrilateral are the vertices of a parallelogram. Each side is a midline of a triangle cut off by a diagonal, so it is parallel to that diagonal and half as long.
Drag the coloured points and watch the numbers under the figure.