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GCD{$\ \cdot\ $}LCM = PROD for Spokes:

If {$T\in\mathcal {B}^+$} is a spoke, then {$$\text{GCD}_{_T}\ \cdot\ \text{LCM}_{_T} = \text{PROD}_{_T}.$$}

Proof:

Consider a spoke, {$T$}, with {$a_{_T}<b_{_T}=c_{_T}$}. From {$\Delta(\langle a_{_T},b_{_T},c_{_T}\rangle)=\langle a_{_T},b_{_T},-a_{_T}\rangle$}, we see that {$x_{_T}=a_{_T}$}, {$y_{_T}=b_{_T}$}, and {$z_{_T}=b_{_T}-a_{_T}$}, so {$\text{GCD}_{_T}=(b_{_T}-a_{_T},a_{_T})$}.

True for {$T'$} Claim: We claim that GCD{$\ \cdot\ $}LCM = PROD holds for the box {$T'=\langle a_{_T},b_{_T},c_{_T}\rangle$}.

We now proceed by induction. We assume that the proposition holds for {$T=\langle r,s,t\rangle$}, that is, {$$\text{GCD}_{_T}\cdot\text{LCM}_{_T}=\text{PROD}_{_T}.$$} We will show that the proposition holds for the three {$\Delta$} pre-images, {$\langle r+s+t,s,t\rangle$}, {$\langle r,s+r+t,t\rangle$}, and {$\langle r,s,t+r+s\rangle$}.

First, we define for any path {$P$} the path vector, {$P=(p_a,p_b,p_c)$}.

The next Claim is that all we need to take the induction step is a set of three statements.

The Path Vector Claim: Let {$P^1$} and {$P^3$} be paths on {$T=\langle r,s,t\rangle$} corresponding to {$L_1$} and {$L_3$}. If

• {$\text{GCD}_{_T}\cdot(p^1_a,p^1_a+p^3_a)=(s+t)$}
• {$\text{GCD}_{_T}\cdot(p^1_b,p^1_b+p^3_b)=(r+t).$}
• {$\text{GCD}_{_T}\cdot(p^1_c,p^1_c+p^3_c)=(r+s).$}

are true, then GCD{$\ \cdot\ $}LCM = PROD holds for the three pre-images.

All we need now is to show inductively that the three statements are true. The next two lemmas do this:

Also True for {$T'$} Claim: Let {$P^1$} and {$P^3$} be paths on {$T'=\langle a_{_T},b_{_T},c_{_T}\rangle$} corresponding to {$L_1$} and {$L_3$}. Then

• {$\text{GCD}_{_{T'}}\cdot(p^1_a,p^1_a+p^3_a)=(b_{_T}+c_{_T})$}
• {$\text{GCD}_{_{T'}}\cdot(p^1_b,p^1_b+p^3_b)=(a_{_T}+c_{_T}).$}
• {$\text{GCD}_{_{T'}}\cdot(p^1_c,p^1_c+p^3_c)=(a_{_T}+b_{_T}).$}

And True Inductively Claim: Let {$P^1$} and {$P^3$} be paths on {$T=\langle r,s,t\rangle$} corresponding to {$L_1$} and {$L_3$}. If the statements

• {$\text{GCD}_{_T}\cdot(p^1_a,p^1_a+p^3_a)=(s+t)$}
• {$\text{GCD}_{_T}\cdot(p^1_b,p^1_b+p^3_b)=(r+t).$}
• {$\text{GCD}_{_T}\cdot(p^1_c,p^1_c+p^3_c)=(r+s).$}

are true on {$T$}, then they are true on all {$\Delta$} pre-images of {$T$}.