I ploted 3D the function `Sin[A/2]Sin[B/2]Sin[C/2]` with `A, B, C > 0` and `A + B + C = Pi`.

[**A basic approach** in the post _How to plot ternary density plots_](https://mathematica.stackexchange.com/a/39751/1871) answers with the use of `FindGeometricTransform`. How can I transform the `Plot3D`ed function inside the equilateral triangle with `FindGeometricTransform`? If there is a simpler method, I would like to know it.

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An example of ternary 3D plot of a geometric inequality

Let ABC be a triangle and its radius of circumcircle R = 1.

In ternary 3D plot, based on A, B, C > 0 and A + B + C = Pi, the value of a function is plotted 3-dimensionally on the equilateral domain.

Denote:

a, b, c = the sidelengths of ABC respectively.

s = (a + b + c)/2.

r = the radii of incenter.

ha, hb, hc = the altitudes of ABC respectively.

The bottom (yellow), middle (magenta) and top (cyan) surfaces show the values of left-hand-side, middle and right-hand-side functions of the following inequality by Mr. George Apostolopouls.

Ga19Mar30: 6rs/R ≤ Sin[A](hb + hc) + Sin[B](hc + ha) + Sin[C](ha + ha) ≤ 3s.
[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/wBSrj.jpg