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2 Eliminated higher dimensions, as calculation was negative.

# Is there a generalized Feuerbach point for an irregular non-Euclidean simplextriangle?

Is the sphere circle externally tangent to the $d+1$ escribed spheres three excircles of an irregular $d$-dimensional non-Euclidean simplex triangle internally tangent to the inscribed sphere incircle of the simplextriangle, the tangent point being a generalized Feuerbach point? In Euclidean plane geometry, the circle externally tangent to the excircles of an irregular triangle is internally tangent to the incircle at the Feuerbach point. My calculations indicate that this should similarly obtain for a non-Euclidean simplextriangle, but I have found no proof. (It is possible that my calculations could, with difficulty, be turned into an analytic proof.) An inscribed sphere is tangent to each facet of a simplex, and its center is on the same side of the facet as the vertex opposite that facet. An escribed sphere is tangent to each facet of a simplex, and its center is on the same side of each facet as the vertex opposite that facet save for one facet where it is on the opposite side. A sphere circle is internally (externally) tangent to another sphere circle if the distance between their centers is the difference (sum) of their radii. A simplex triangle is regular irregular if all its edges are of it is not equilateral.

In response to a helpful comment, my calculations indicate that the same lengthadditional tangency does not obtain in higher dimensions, and otherwise irregular. We restrict non-Euclidean so I have revised the question to spherical and hyperbolic spacesapply to only a two-dimensional space.

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# Is there a generalized Feuerbach point for an irregular non-Euclidean simplex?

Is the sphere externally tangent to the $d+1$ escribed spheres of an irregular $d$-dimensional non-Euclidean simplex internally tangent to the inscribed sphere of the simplex, the tangent point being a generalized Feuerbach point? In Euclidean plane geometry, the circle externally tangent to the excircles of an irregular triangle is internally tangent to the incircle at the Feuerbach point. My calculations indicate that this should similarly obtain for a non-Euclidean simplex, but I have found no proof. (It is possible that my calculations could, with difficulty, be turned into an analytic proof.) An inscribed sphere is tangent to each facet of a simplex, and its center is on the same side of the facet as the vertex opposite that facet. An escribed sphere is tangent to each facet of a simplex, and its center is on the same side of each facet as the vertex opposite that facet save for one facet where it is on the opposite side. A sphere is internally (externally) tangent to another sphere if the distance between their centers is the difference (sum) of their radii. A simplex is regular if all its edges are of the same length, and otherwise irregular. We restrict non-Euclidean to spherical and hyperbolic spaces.