X is always inside one of the circles, and outside the other, except for the degenerate case where X is at M or N. So with respect to one circle, the line pair EX YZ represents intersecting chords, while with respect to the other, the pair represents intersecting secants.
The relevant line segments are:
XY, EX, MX, NX, FM, NZ
From intersecting chords:
From intersecting secants:
c*(c+e) = d*(d+f)
a*d=b*c=c*(c+e)=d*(d+f), hence a=d+f, and the ratio is 1:1.
Posted by broll
on 2015-03-04 20:58:22