parallelogram and metric dependencies

[Tochka]

Hallo, can you please help me find the solution of this geometry problem:

In the parallelogram ABCD, the heights AD and BF intersect the diagonal AC at points M and N, respectively. Find the sides of the parallelogram if AM = a, MN = b, NC = c.

Thank you in advance!

 

[Dr.Peterson]

Please check that you have copied the problem correctly; a picture may be needed.

I wouldn't call the side AD a "height" (altitude), and if you meant AE (E being the foot of the perpendicular to CD), still the intersection of AE with AC is A, not another point. So I'm going to guess it meant DE, with E being on AB (or BC?).

But I won't pursue it without (a) confirmation of the actual problem, and (b) evidence of your own thinking, so I can know where you need help.

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[HallsofIvy]

In particular, AD is a side of the parallelogram ABCD, not a "height" (unless this is a rectangle).

 

[Dr.Peterson]

If I suppose that AD was meant to be DE, and that the altitudes are both perpendicular to sides AD and BC (there are other choices), then it might look like this:



Again, I won't make any attempt until I know I have it right. It isn't immediately clear whether the three known lengths (really only two) would be sufficient.