Problem with 3 cones - wanted a symbolic solution

[ValeryOchkov]

Here we have tried to find a symbolic solution for the problem of two cones.

But we can cut out three cones from a round billet!

We can show with the help of graphics and numerical mathematics that the third cone is superfluous - the maximum total volume can be only for two cones.
Can we show this using symbolic MATHEMATICS?

See please pics and Mathcad Prime 5.0 sheet.

Fig.1Fig.2Fig.3Fig.4Fig.5Fig.6Fig.7Fig.8Fig.9Fig.10

[ValeryOchkov]

Here must be the triangle diagram

Fig. 11Fig. 12