$\bold\large{C}$

$\bold\large{B}$

$\bold\large{A}$

$\textbf{Reference Triangle}$

$\bold\Large{c}$

$\bold\Large{a}$

$\bold\Large{b}$

$\bold\large{X_{1144}}$

$\bold{X_{1}}$

$\bold{X_{6}}$

$\textbf{Circumhyperbola(\textit{A,B,C,X}_{1},\textit{X}_{6})}$

$\textbf\textsc{Triangle Center}$$\bold{\large\color{0,140,0}{X_{1144} }}$$\textbf{\textsc{- Ehrmann Congruent Squares Point of the Reference Triangle \large\textit\color{200,0,0}{ABC}}}$

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$\bold{X_{371}}$

$\bold{X_{372}}$

Hide Conic

C: (6, 0)
B: (0, 0)
A = (9.926721961102732, 8.453737288549378)
ref3gon, c, a, b = Polygon(A, B, C)
tAng_1, tAng_2, tAng_3 = Angle(ref3gon)
S_{2Δ}: (2 * Area[ref3gon])
k_3(S) = a^(2) + b^(2) + c^(2) + S
p_{2s} = Perimeter(ref3gon)
k_2(m, S) = (m * (a^(3) + b^(3) + c^(3) - (p_{2s} * k_3(S))))
abc: ((a * b) * c)
k_1(S) = (abc * p_{2s}) + (((a * b) + (b * c) + (c * a)) * S)
f_L(m, S, L) = (k_3(S) * L^(3)) + (k_2(m, S) * L^(2)) + (k_1(S) * L) - (abc * S_{2Δ})
f_{1144}(L) = f_L(1, S_{2Δ}, L)
r1144_{list}: x({Root[f_{1144}]})
L_{E} = Element(r1144_{list}, 1)
x_{1144} = Trilinear(A, B, C, a / (a - L_{E}), b / (b - L_{E}), c / (c - L_{E}))
x_{1} = TriangleCenter(A, B, C, 1)
x_{6} = TriangleCenter(A, B, C, 6)
ch = Conic(A, x_{1}, x_{6}, C, B)
tx = Text(Distance[x_{1144}, Line[B, C]])
ty = Text(Distance[x_{1144}, Line[C, A]])
tz = Text(Distance[x_{1144}, Line[A, B]])
axesOn = true
infoOn = false
x_{371} = TriangleCenter(A, B, C, 371)
x_{372} = TriangleCenter(A, B, C, 372)
TP_A = Segment(A, x_{1144})
TP_B = Segment(B, x_{1144})
TP_C = Segment(C, x_{1144})
CP_1 = Intersect(TP_C, Line[Intersect[Circle[B, L_{E}], a], TP_B])
CP_2 = ClosestPoint(a, CP_1)
AP_1 = Intersect(TP_A, Line[Intersect[Circle[C, L_{E}], b], TP_C])
AP_2 = ClosestPoint(b, AP_1)
CS_A = If(y(A) ≥ 0 ∧ x(C) ≥ 0 ∨ y(A) < 0 ∧ x(C) < 0, Polygon[CP_2, CP_1, 4], Polygon[CP_1, CP_2, 4])
CS_B = If(y(A) ≥ 0 ∧ x(C) ≥ 0 ∨ y(A) < 0 ∧ x(C) < 0, Polygon[AP_2, AP_1, 4], Polygon[AP_1, AP_2, 4])
BP_1 = Intersect(TP_B, Line[Intersect[Circle[A, L_{E}], c], TP_A])
BP_2 = ClosestPoint(c, BP_1)
CS_C = If(y(A) ≥ 0 ∧ x(C) ≥ 0 ∨ y(A) < 0 ∧ x(C) < 0, Polygon[BP_2, BP_1, 4], Polygon[BP_1, BP_2, 4])
conicOn = true
cy = Text(y(x_{1144}))
aA = Text(tAng_1)
cx = Text(x(x_{1144}))
aB = Text(tAng_2)
aC = Text(tAng_3)
text1: "$\text{\textbf{Sidelengths} of \bf\it\color{200,0,0}{ABC}}$
$\text{Side \textbf{\it{a}} = \bf{" + (LaTeX[a]) + "}}$
$\text{Side \textbf{\it{b}} = \bf{" + (LaTeX[b]) + "}}$
$\text{Side \textbf{{\it{c}} = \bf{" + (LaTeX[c]) + "}}$"
text2: "$\text{\textbf{Vertex Angles} of \bf\it\color{200,0,0}{ABC}}$
$\text{∡\textbf{\it{A}} = \bf{" + (LaTeX[aA]) + "}}$
$\text{∡\textbf{\it{B}} = \bf{" + (LaTeX[aB]) + "}}$
$\text{∡\textbf{{\it{C}} = \bf{" + (LaTeX[aC]) + "}}$"
text3: "$\text{\textbf{Trilinear Coordinates} of \bf\color{0,140,0}{\textit{X}_{1144}}}$
$\text{\bf{\it{x}} = \bf{" + (LaTeX[tx]) + "}}$
$\text{\bf{\it{y}} = \bf{" + (LaTeX[ty]) + "}}$
$\text{\bf{\it{z}} = \bf{" + (LaTeX[tz]) + "}}$"
text4: "$\text{\textbf{Cartesian Coordinates} of \bf\color{0,140,0}{\textit{X}_{1144}}}$
$\text{\bf{\it{x}} = \bf{" + (LaTeX[cx]) + "}}$
$\text{\bf{\it{y}} = \bf{" + (LaTeX[cy]) + "}}$"
showInfo: true
text0: "  $\text{Sidelength of Ehrmann Congruent Squares}$$\bold\color{0,140,0}{L_\text{E}}$$\textbf{= " + (LaTeX[L_{E}]) + "}$"