alexb,

Thanks for making that applet on the parabola, and noticing that M and K also lie on CD. This whole thing is just a projection of the same situation in a circle, which is how the idea came to me in the first place(while looking at Archimedes Book of Lemmas, Proposition 12). I have since realized, though, that one should be very wary of parabolas in composing problems, since generally it's so easy to deal with the equations y=x^2 and y'=2x, and that usually one can just mess around with equations to prove almost anything true. This problem is not too hard if you just write down equations and crank. Oh well.

As for the second one, probably it's no surprise that I noticed that while reading about Sangaku. It's roughly equivalent to an old Sangaku problem.

Greg