Successive lines parallel to the sides of a triangle

Let ABC be a triangle in the plane and M0 a point on the line B C and:
- the line parallel to the line A B through M0 cuts the line C A in a point M1,
- the line parallel to the line B C through M1 cuts the line A B in a point M2,
- the line parallel to the line C A through M2 cuts ... and so on (see the figure below).
Then the points M0, M3 and M6 are on the line B C, the points M1 and M4 on the line C A, and M2, M5 on the line A B.
The following segments are parallel: M0M1 || M3M4 || AB,   M1M2 || M4M5 || BC,   M2M3 || M5M6 || CA.

Three problems

1. Show that M6 = M0.
2. Let A¢ be the midpoint of the segment BC. Define 
x : = 
A¢M0
A¢C
.
a) Is it possible to have M0 = M1 and if yes, for which values of x ?
b) Is it possible to have M0 = M2 and if yes, for which values of x ?
c) Is it possible to have M0 = M3 and if yes, for which values of x ?
d) Is it possible to have M0 = M4 and if yes, for which values of x ?
e) Is it possible to have M0 = M5 and if yes, for which values of x ?
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Instructions

In the Sketch the red points are draggable. You may push the buttons on the left upper corner of the Sketch.
3. Algebraic (signed) area. If a line turns around a piece of plane in the positive direction (counter-clockwise), the area is counted positively. It is counted negatively if you turn in the negative direction. 
Let S be the algebraic area of ABC. Let us denote the area enclosed in M0 M1 M2 M3 M4 M5 M0 by Sf(x).
Find f(x) for
x = 1,     x > 1,     x = 1/3,     1/3 < x < 1,    x = 0,     0 < x < 1/3.
Eric Lehman & Martti E. Pesonen 2007