Inside a convex quadrilateral ABCD there are points P (and P', P'' etc))such that drawing WP,XP,YP,ZP one gets four
quadrilaterals of equal areas (W, X, Y, Z being the midpoints of the sides).
Define P's location.
Say W is on AB, X is on BC, Y is on CD, and Z is on AD.
If W, X, Y, Z are the midpoints of the sides then WXYZ is the Varignon parallelogram. https://en.wikipedia.org/wiki/Varignon%27s_theorem
Construct lines through A,B,C, and D parallel to WZ, WX, XY, and YZ. These lines intersect to form a new parallelogram, STUV.
The diagonals of STUV intersect at P.
Even if the quadrilateral is concave, or crossed.
Posted by broll
on 2018-04-28 23:55:50