It's a little problem that I cannot disprove or prove .
We work with the following picture :
First we draw a circle and his tangent at point $D$ and $E$ .Secondly we draw the radius or the line $DE$.Thirdly we draw the line $EH$ and we get the points $I$ and $H$.Then we draw the parallel to the tangent of the circle wich goes at the point $I$ . And finally draw the perpendicular to the tangent of the circle wich goes to the point $H$ .We get the point $M$ wich is a point of the curve named Witch of Agnesi . Now we do the same things with the circle in red and in violet (always with the line $EH$).
Prove that $M$ ,$N$ , $R$ are aligned (line orange).
I don't find any counter-example so I was thinking to a proof . I have tried to prove that $\angle MRN=180^°$ and use the fact that we have many right triangles.Notably I think that the line $MR$ goes through the intersection of the tangent of the red circle and the blue circle (same thing for the violet tangent and the red circle) .Finally I have tried to use Pappus's hexagon theorem without success. I think it's not hard but I need help on this question.
Thanks a lot for all your contributions.
Ps:The circle in red goes through the center of the blue circle (same things for the violet circle and the center of the red circle)
One way to prove this is with a use of analytic geometry.
Let $E=(0,0)$ and $D = (0,4a)$ and $$EH: \;\;y=kx\;\;$$ so $H=({4a\over k},4a)$. Notice that $DI\bot EH$ so the equation of $$DI:\;\; y= -{1\over k}x +4a$$
so $$I = \Big({4ak\over k^2+1}, {4ak^2\over k^2+1}\Big)$$ and thus
$$M = \Big({4a\over k}, {4ak^2\over k^2+1}\Big)$$ so we see that $M$ is on a line $$y= {k^3\over k^2+1}x$$
Similary we see that $N$ and $R$ are also on this line.
Notice that $F$ and $O$ can be choosen arbitrary on $DE$, need not to be the centers of circles.