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X₃₈₆₂–X₄₀₀₀¶

Inverse Central Triangle Combos Xcom[Inverse(n(T))]

Perspectors of ABC and parameterized central triangles T(f, g) (X₃₈₆₂–X₃₈₆₇), inverse 3rd/4th/5th Euler triangle combos (X₃₈₆₈–X₃₉₁₀), and combos and inverse-triangle combos across the intouch, extouch, orthic, tangential, hexyl, Brocard, incentral, excentral, intangents, and extangents triangles (X₃₉₁₁–X₄₀₀₀).

X₃₈₆₂–X₄₀₀₀ (Inverse Central Triangle Combos Xcom[Inverse(n(T))]) — 16 representative centers (from 139 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

Group Preamble — Inverse Triangles and More Combos

The Euler triangle and the 2nd, 3rd, 4th, and 5th Euler triangles are discussed in the preamble to X₃₇₅₈, along with eight other central triangles using the notation T(f(a,b,c), g(b,c,a)). Recall that the inverse of a normalized central triangle T, denoted by Inverse(nT) or Inverse(n(T)), is also a normalized central triangle. Barycentrics for the inverse of each of the 13 triangles and properties of these triangles are given (Peter Moses, December 2011) as follows:

Inverse(n(Euler triangle)) = T(f(a,b,c), g(b,c,a)), where
\(f(a,b,c) = (b^{2} - c^{2})^{2} + a^{2}(3a^{2} - 4b^{2} - 4c^{2})\)
\(g(b,c,a) = b^{4} - (c^{2} - a^{2})^{2}\)

Inverse(n(2nd Euler triangle)) = T(f(a,b,c), g(b,c,a)), where
\(f(a,b,c) = a^{2}(a^{2} - b^{2} - c^{2})(a^{6} + b^{6} + c^{6} - a^{4}b^{2} - a^{4}c^{2} - a^{2}b^{4} - a^{2}c^{4} - 2a^{2}b^{2}c^{2} - b^{4}c^{2} - b^{2}c^{4})\)
\(g(b,c,a) = b^{2}(b^{2} - c^{2} - a^{2}(a^{6} + b^{6} - c^{6} - a^{4}b^{2} - 3a^{4}c^{2} - a^{2}b^{4} + 3a^{2}c^{4} + 2a^{2}b^{2}c^{2} - 3b^{4}c^{2} + 3b^{2}c^{4})\)

Inverse(n(3rd Euler triangle)) = T(f(a,b,c), g(b,c,a)), where
\(f(a,b,c) = a(b^{2} + c^{2} - ab - ac + bc)\)
\(g(b,c,a) = b(a^{2} - ab - ac + bc - c^{2})\)

Inverse(n(4th Euler triangle)) = T(f(a,b,c), g(b,c,a)), where
\(f(a,b,c) = a(a - b - c)(b^{2} + c^{2} + ab + ac)\)
\(g(b,c,a) = b(-a + b + c)(a^{2} + ab - ac - bc - c^{2})\)

Inverse(n(5th Euler triangle)) = T(f(a,b,c), g(b,c,a)), where
\(f(a,b,c) = (a^{2} - 2b^{2} - 2c^{2})(3a^{2} + b^{2} + c^{2})\)
\(g(b,c,a) = -(a^{2} - b^{2} - c^{2})(2a^{2} - b^{2} + 2c^{2})\)

Inverse(n(T(bc,\(b^{2})\)) = T(f(a,b,c), g(b,c,a)), where
\(f(a,b,c) = -bc(a^{2} - bc)(b^{2} + c^{2} + bc)\)
\(g(b,c,a) = b^{2}(ab - c^{2})(a^{2} + c^{2} + ac)\)

Inverse(n(T(-bc,\(b^{2})\)) = T(f(a,b,c), g(b,c,a)), where
\(f(a,b,c) = bc(a^{2} - bc)(b^{2} + c^{2} - bc)\)
\(g(b,c,a) = b^{2}(ab + c^{2})(a^{2} + c^{2} - ac)\)

Inverse(n(T(bc,\(c^{2})\)) = T(f(a,b,c), g(b,c,a)), where
\(f(a,b,c) = -a(a^{2} - bc)(b^{2} + c^{2} + bc)\)
\(g(b,c,a) = (ab - c^{2})(a^{2} + c^{2} + ac)\)

Inverse(n(T(-bc,\(c^{2})\)) = T(f(a,b,c), g(b,c,a)), where
\(f(a,b,c) = -a(a^{2} - bc)(b^{2} + c^{2} - bc)\)
\(g(b,c,a) = (ab + c^{2})(a^{2} + c^{2} - ac)\)

Inverse(n(\(T(a^{2}\),\(a^{2}\) - \(b^{2})\)) = T(f(a,b,c), g(b,c,a)), where
\(f(a,b,c) = (3a^{2} - b^{2} - c^{2})(b^{4} + c^{4} - b^{2}c^{2})\)
\(g(b,c,a) = (a^{2} - 3b^{2} + c^{2})(a^{2}b^{2} - 2b^{2}c^{2} + c^{4})\)

Inverse(n(\(T(a^{2}\),\(a^{2}\) - \(c^{2})\)) = T(f(a,b,c), g(b,c,a)), where
\(f(a,b,c) = a^{2}(b^{2} + c^{2} - 3a^{2})\)
\(g(b,c,a) =(a^{2} - c^{2})(a^{2} - 3b^{2} + c^{2})\)

Inverse(n(\(T(a^{2}\),\(a^{2}\) + \(b^{2})\)) = T(f(a,b,c), g(b,c,a)), where
\(f(a,b,c) = -(3a^{2} + b^{2} + c^{2})(b^{4} + c^{4} + b^{2}c^{2})\)
\(g(b,c,a) = (a^{2} + 3b^{2} + c^{2})(a^{2}b^{2} + c^{4})\)

Inverse(n(\(T(a^{2}\),\(a^{2}\) + \(c^{2})\)) = T(f(a,b,c), g(b,c,a)), where
\(f(a,b,c) = -a^{2}(a^{2} + b^{2} + c^{2})(3a^{2} + b^{2} + c^{2})\)
\(g(b,c,a) = (a^{2} + c^{2})(a^{2} + b^{2} - c^{2})(a^{2} + 3b^{2} + c^{2})\)

For present purposes, label these last triangles as IT₁, IT₂, IT₃, IT₄, IT₅, IT₆, IT₇, IT₈; then all except IT₅ and IT₆ are perspective to the reference triangle ABC; perspectors are described at X₃₈₆₂-X₃₈₆₆. Label the corresponding original triangles T₁, T₂, T₃, T₄, T₅, T₆, T₇, T₈; then the following pairs are perspective: (T₁, IT₁), (T₂, IT₂), (T₃,IT₃), (T₄, IT₄), (T₇, IT₇), (T₈,IT₈); coordinates for the perspectors are long and omitted.

The triangles IT₁ and IT₂ are triply perspective to ABC (as are T₁ and T₃). Order-label IT₁ as A'B'C'. The three perspectivities are then given by

AA'∩BB'∩CC' = X3862
AB'∩BC'∩CA' =\(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = ab(a^{2} - bc)(c^{2} - ab)(b^{2} + bc + c^{2})\)
AC'∩BA'∩CB' =\(f(a,c,b) : f(b,a,c) : f(c,b,a)\)

Note that the second two perspectors are a bicentric pair (not triangle centers; see TCCT, page 47); likewise for the other triple perspectivity, with IT₃ order-labeled as A'B'C':

AA'∩BB'∩CC' = X3864
AB'∩BC'∩CA' =\(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a(a^{2} - bc)(c^{2} - ab)(b^{2} + bc + c^{2})\)
AC'∩BA'∩CB' =\(g(a,c,b) : g(b,a,c) : g(c,b,a)\)