MHB Proving Nine-Point Circle Theorem w/ Parallelogram & Symmetry

[pholee95]

Hi, I'm stuck on this problem and would like some help.

The purpose of this exercise is to prove the Nine-Point Circle Theorem. Let triangleABC be
a Euclidean triangle and let points D, E, F, L, M, N, and H be as in Figure 8.46. Let γ
be the circumscribed circle for triangleDEF.

a) Prove that quadrilateralEDBF is a parallelogram. Prove that DB=DN. Use a symmetry
argument to show that N lies on γ. Prove, in a similar way, that L and M lie on γ.

I have attached on how the picture looks like.

 

[Greg]

In general EDBF is not a parallelogram. You need to be given some extra information about the circle. One may expect that this information would be that points L, M and N are on the circle but you're expected to prove that they are in a subsequent part of the exercise. This leads me to ask if you've typed the problem accurately.

If you have a point of clarification, that's good, but please post some work and/or thoughts on how to approach the problem as well.