Geometric configuration with lots of properties

Question

See.
To sum up: ABCD lie on a circle m with center M. BDEF lie on a circle n with center N. ABE,ADF,BCF,CDE are collinear.
(So when constructing, draw the two circles m,n, find their intersections B,D, choose some C on m, find the intersection E,F of BC and CD with n, find the intersection A of BE and DF, and pray that A lies on m :-)
My bet is this works when...
a) MBN=MDN=right angle
b) EGF=right angle
c) MGN are collinear
d) You can prove it fastest with vectors.

Answer 1

This construction works if and only if the two circles are orthogonal to each other, i.e. MBN = MDN = right angle is the necessary and sufficient condition for this construction to always work. The rest is wrong. MGN are collinear iff G is the midpoint of BD and I dont know what G is according to you. EGF is definitely not right. d) is laughable at best. I just stared at a couple of pictures for five to ten minutes and I got all the proofs without a single line of computation.