A piece of paper has the precise shape of a triangle (which we will denote by triangle ABC), where the respective lengths of the crease whenever the paper is folded such that; the vertex A is joined onto vertex B, the vertex A is joined onto vertex C and, the vertex B is joined onto vertex C are 7, 8 and 9.

Determine the lengths AB, AC and BC.

(In reply to 6 solutions (unverified) by Jer)

In my solution I assumed that it didn't matter the length of the vertical crease so I made it 8.  This yields the solutions with the side lengths I gave:

[15.0187, 13.1414, 11.3183]  <--Bractals found (works)
[127.9576, 111.9629, 21.8458] <-- works

If the vertical crease is 7 we get:

[7.4448, 8.5083, 8.2666]  <---- This doesn't work creases 7, 5.42, 5.58 because the crease intersect the wrong sides
[279.7167, 249.5462, 34.5880] <---- This doesn't work in two ways creases 8.97, 8, 9 or 7.64, 7, 8 again with creases intersecting wrong. 

If the vertical crease is 9 we get:

[18.7771, 14.6044, 13.0363]  <-- Bractals found (works)
[99.3676, 77.859, 27.0844] <-- works

It occurs to me we may get additional solutions if the x-coordinates of both points are positive.  I'll look into that.

Posted by Jer on 2012-05-22 14:06:47