Here is a problem that is driving me crazy. There ought to be a simple solution, but it eludes me.

I've made a model from paper, and it is a feasible construction, that constrains h and point S:

OX, OY, OZ are orthogonal XYZ axes.

O is a fixed point. T lies on OX, R lies on OY, and P lies on OZ.

S lies in the XY plane, so base kite ROTS is a planar quadrilateral (i.e. there is no hinge along OS).

a, b, c, d & e are known (of fixed length).

For convenience, TO > OR, and d > e.

Also, c > b > a.

What then is h (=OP) ???

I can imagine raising point P along OZ, so points R, S and T on the base plane draw closer to O and lie in the XY plane thru O, while keeping a, b, c, d & e of fixed length(s), with R lying somewhere along OY, and T somewhere along OX.

But I cannot see how to calculate h, (and thus solve the base kite dimensions and angles, as well as the rest of the pyramid).

Yet all the points ought to be uniquely determined for the solution. My paper model works just fine, but I want a precise solution.

Any clues to solve h, and thus the rest of the unknowns? I attach a pdf and a jpg of the construction (MyProblem.pdf and MyProblem.jpg).

Your assistance would be greatly appreciated!

Regards, Rob.