Edited by Richard Walker, Friday, 28 June 2024, 16:02
Varignon's Theorom says that if we take any quadrilateral whatsoever and join the midpoints of its sides we (rather surprisingly) always get a parallelogram
I give a neat and I hope fairly intuitive proof here.
Today I read about an elegant generalisation of this theorem. The page I've linked to has much more that this, but in my diagram below, which I drew using Geogebra, I've just illustrated the first case.
For any hexagon, STUVWX in the figure, form a triangle from each group of three adjacent vertices in turn. Mark the centroid, i.e. the centre of gravity, of each triangle, and join them up to form a hexagon A1B1C1D1E1F1 as shown.The each pair of opposite sides of the new hexagon are parallel and of equal length, in other words the new hexagon is analogous to a parallelogram, but with six sides rather than four.
Varignon and beyond
Varignon's Theorom says that if we take any quadrilateral whatsoever and join the midpoints of its sides we (rather surprisingly) always get a parallelogram
I give a neat and I hope fairly intuitive proof here.
Today I read about an elegant generalisation of this theorem. The page I've linked to has much more that this, but in my diagram below, which I drew using Geogebra, I've just illustrated the first case.
For any hexagon, STUVWX in the figure, form a triangle from each group of three adjacent vertices in turn. Mark the centroid, i.e. the centre of gravity, of each triangle, and join them up to form a hexagon A1B1C1D1E1F1 as shown.The each pair of opposite sides of the new hexagon are parallel and of equal length, in other words the new hexagon is analogous to a parallelogram, but with six sides rather than four.