X₁₃₁₄

3rd MOSES INTERSECTION

X₁₃₁₄ is a point of intersection of the Euler line and the incircle. For some obtuse triangles, this point is not in the real plane (specifically, for those a,b,c such that Q < 0).

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₃₁₄ (3rd MOSES INTERSECTION) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER)

Trilinears: \(bcf(a,b,c) : caf(b,c,a): abf(c,a,b)\), where f(a,b,c) is the 1st barycentric given below

Barycentrics: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = [d^{2} + (4r - R)R +\)\(\sqrt{Q}\)]\(S_{B}\)\(S_{C}\) + (\(d^{2}\) + \(2r^{2}\) - \(R^{2})\)\(a^{2}\)\(S_{A}\), where \(Q = 4d^{2}R(4r - R) + [d^{2} - 3R^{2} + 4r(r + R)]^{2}\), d = distance between X₃ and X₄, R = circumradius, r = inradius, \(S_{A} = (b^{2} + c^{2} - a^{2})/2\), and \(S_{B}\) and \(S_{C}\) are defined cyclically (Peter J. C. Moses, 3/2003)

Search Value (6, 9, 13): \(0.871828089890\)

X₁₃₁₄ lies on the incircle.

▸ Incident Central Lines1 line

{2, 3}

Notes

As a point on the Euler line, X₁₃₁₄ has Shinagawa coefficients \((d^{2} + 2r^{2} - R^{2},4rR - 2r^{2} + Q^{1/2})\).