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{$L$}, {$R$}, {$M$}, {$C$} as maps on the box triangle

We can think of the box triangle as the set of all triples of non-negative reals, {$\langle a,b,c\rangle$}, with {$a+b+c>0$}, modulo the equivalence relation: {$$\langle a,b,c\rangle\sim\langle a',b',c'\rangle\ \text{iff one is a multiple of the other.}$$} It should be clear that all four functions preserve equivalence, e.g., {$L(rT)=rL(T)$}.

If we think instead of the the box triangle as the set of all triples {$\langle a,b,c\rangle$}, with {$a+b+c=1$}, then we would write: {$$L(a,b,c)=(\frac 1{1+b+c},\frac b{1+b+c},\frac c{1+b+c})$$} {$$R(a,b,c)=(\frac a{1+b+c},\frac b{1+a+b},\frac 1{1+b+c})$$} {$$M(a,b,c)=(\frac a{1+a+c},\frac 1{1+b+c},\frac c{1+a+c}c)$$} {$$C(a,b,c)=(\frac{b+c}2,\frac{a+c}2,\frac{a+b}2)$$}