Place the square with coordinates
A=(0,0)
B=(1,0)
C=(1,1)
D=(0,1)
rotate by angle x so that
A=(0,0)
B'=(cosx,sinx)
C'=(cosxsinx,cosx+sinx)
D'=(sinx,cosx)
call the midpoints:
of BD=(1/2,1/2) = E
of B'D=(cos(x)/2,(1+sin(x))/2) = F
of B'D'=((cos(x)sin(x))/2,(cos(x)+sin(x))/2) = G
of BD'=((1sin(x))/2,cos(x)/2) = H
To show a quadrilateral is a square we can show its consecutive sides are perpendicular and its diagonals are perpendicular.
Slope EF = Slope GH = sin(x)/(cos(x)1)
Slope FG = Slope HE = (1cos(x),sin(x))
Slope EG = (cos(x)+sin(x)1)/(cos(x)sin(x)1)
Slope FH = (cos(x)+sin(x)+1)/(cos(x)+sin(x)1)
QED

Posted by Jer
on 20130927 10:16:15 