Existence of the Nine-Point Circle Observe the figure for this problem, where ∆ABC is a gi...

Existence of the Nine-Point Circle Observe the figure for this problem, where ∆ABC is a given triangle, L, M, N are the midpoints of the sides, D, E, and F are the feet of the three altitudes on the sides, H is the orthocenter, and X, Y, and Z are the midpoints of segments and . Is there any reason you can think of that would make quadrilateral MNYZ a parallelogram? A rectangle? In the middle figure in the sequence, the diagonals are drawn, intersecting at point U, Show that a circle centered at U passes through M, N, Y, and Z. Finally, use the third figure in the sequence to show that a circle centered at U' passes through M, N, Y, Z, X, and L. (Why does U' – U?) For what reason do points D, E. and F lie on this circle?