Two circles C₁ and C₂ intersect at M and N. A line through M intersects C₁ and C₂ again at E and F, respectively, and X is the mid-point of EF.

The line through N and X intersects C₁ and C₂ again respectively at Y and Z.

Find the ratio YX:XZ

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
XY=a
EX=b
MX=c
NX=d
FM=e
NZ=f

From intersecting chords:
a*d=b*c
From intersecting secants:
c*(c+e) = d*(d+f)
Given: b=c+e
a*d=b*c=c*(c+e)=d*(d+f), hence a=d+f, and the ratio is 1:1.

Nice problem.

Posted by broll on 2015-03-04 20:58:22