Excentral Invariants
Consider the excentral triangle \(A'B'C'\) [25], tangent to the ellipse at the orbits’ vertices \(ABC\).
Since \(A'=\frac{\pi-A}{2}\), \(B'=\frac{\pi-B}{2}\), and \(C'=\frac{\pi-C}{2}\), the family of orbit excentral triangles will only contain acute triangles. Direct derivation produces:
\[
\begin{array}{rcll}
-\cos(2A')-\cos(2B')-\cos(2C') & = & \\ \cos(A)+\cos(B)+\cos(C) & = & 1+\frac{r}{R} & = \text{invariant}
\end{array}
\]
Furthermore, since the orbit will be the excentral’s orthic triangle [25], (i) the excentral’s circumradius \(R'\) will be twice the orbit’s (\(R'=2R\)), and (ii) the orbit’s inradius \(r\) will be given by the following relation [26]:
\[
\begin{array}{rcl}
r & = & 2 R'\;|\cos(A')\cos(B')\cos(C')| = \\
& = & 4 R\;\cos(A')\cos(B')\cos(C')
\end{array}
\]
In the above we removed the absolute value since \(A'\), \(B'\), and \(C'\) are acute. Since \(r/R\) of the orbit is an invariant, the family of excentral triangles will conserve the product of its internal cosines:
\[
\cos(A')\cos(B')\cos(C') = \frac{r}{4R} = \text{constant}
\]
The above relation can be visualized below, where the orbit (resp. excentral) \((u,v)\) triangle is shown blue (resp. green), along with the locus of \((u,v)\). For the orbits (resp. excentrals), this is an iso-curve of cosine sum (resp. product), shown as a dotted blue (resp. green) curve. Cosine sum (resp. product) are shown as the left (resp. right) pictures below and they can be seen dynamically on this video:
Furthermore since (i) the area of any triangle is given by the product of its inradius by its semiperimeter, and (ii) by the product of its circumradius by its orthic semiperimeter [25], we can state, about the excentral triangle:
\[
\begin{array}{rclll}
A' & = & r'\,P'/2 & = & R'\,P/2 \\
P'\frac{r'}{R'} & = & P & = & \text{constant}
\end{array}
\]
Theorem 3: The inradius-to-circumradius ratio for all orbits is invariant.
- Corollary 3.1 The sum of cosines of orbits’ internal angles is constant and equal to \(1+r/R\).
- Corollary 3.2 The negative of sum of double-angle cosines of the excentral triangle is constant and equal to \(1+r/R\).
- Corollary 3.3 The product of cosines of the excentral triangle is constant and equal to \(r/(4R)\).
- Corollary 3.4 The excentral triangle conserves the product of its perimeter by the ratio of its inradius by circumradius. This quantity is equal to the orbits’ constant perimeter.
A formula derived by Feuerbach states that the ratio of areas between a triangle and its orthic is equal to \(2R_h/r_h\), where the subscripted quantities refer to the orthic [14]. Because the orbits are their excentrals’ orthic, we can state that:
\[
A_{exc}/A_{orbit} = 2R/r = \mbox{constant}
\]
- Corollary 3.5 The excentral-to-orbit area ratio is constant and equal to \(2R/r\).
Orthic Invariants
Extending this analysis to the orbits’ orthic triangles, for acute orbits, orthic angles \(\theta_i'', i=1,2,3\) will be equal to \(\pi-2\theta_i\), i.e., \(\sin(\theta_i''/2)=cos(\theta_i)\). Given constancy of sum of orbit cosines, the orthic of an acute orbit will conserve the sum of half-angle sines.
Likewise, since the area \(A\) of a triangle is \(r\,P/2\) and, if acute, also \(R\,P''/2\) [25], the orthic of an acute orbit will conserve its perimeter \(P''=P\,r/R\), i.e., the product of the two orbit invariants. If the orbit \(T\) is obtuse, two of its orthic vertices will lie outside \(T\) as will \(T\)’s orthocenter \(H\) [3]. Let \(T_a\), be the excentral triangle of the orthic, always acute. The orthic of \(T_a\) is the same as \(T\)’s, i.e., the orthic pre-image contains \(T\) obtuse and \(T_a\) acute:
Generalizing the above argument, the area \(A_a\) of the acute preimage \(T_a\) of the orthic will be \(r_a\,P_a/2\) and, all extriangles are acute [25], also \(R_a\,P''/2\). So for both acute and obtuse orbits, the orthic perimeter \(P''\) will be given by \(P_a\,r_a/R_a\), where \(R_a,r_a,P_a\) are the circumradius, inradius, and perimeter of the acute preimage of the orthic: its extriangle (resp. the orbit) for obtuse (resp. acute) orbits.
The angular quantity the orthic conserves for obtuse orbits is still under investigation.
Summary of Invariants
We can summarize the following characteristics for orbit, its excentral triangle, and its orthic:
\[
\begin{array}{|c|c|c|c|c|}
\hline
\bf{\text{Triangle}} & \bf{\text{All Acute}} & \bf{\text{Some Obtuse}} & \bf{\text{Angular Invariance}} & \bf{\text{Metric Invariance}} \\
\hline
\text{Orbit} & a<1.35 & a>1.35 & \sum{\cos(\theta_i)} = 1+r/R & P \\
\text{Excentral} & \forall a & \varnothing & \prod{\cos(\theta_i')}=r/(4R) & P'\frac{r'}{R'} = P \\
\text{Orthic} & a<1.17 & a>1.17 & \sum{\sin{(\theta_i''/2)}}=1+r/R\;^\dagger & P'' = P_a\frac{r_a}{R_a}\;{^\dagger}{^\dagger} \\
\hline
\end{array}
\]
\(^\dagger\)for acute orbits only.
\({^\dagger}{^\dagger}\)for acute orbits, this reduces to \(P\,r/R\).
