[alogo] Impossible case to inscribe a rectangula hyperbola

Problem: Construct all rectangular hyperbolas tangent to four given lines determined by the sides of a quadrilateral q = A1A2A3A4 (may be convex or not).
The problem admits either two solutions or no solution at all. Since a triangle whose sides are tangent to a rectangular hyperbola must be obtuse-angled, the problem has solutions only if all partial triangles formed out of the four given points are obtuse-angled.
By partial triangles I mean the four triangles formed by selecting three sides of q. The picture below shows an impossible case. Two out of the four partial triangles are not obtuse-angled.

[0_0] [0_1]
[1_0] [1_1]

The picture shows also the conjugate circles of the four partial triangles. Their centers are at the orthocenters of the partial triangles. The parabola tangent to the sides of q is also shown. See the file RectHyperbolasTangent4Lines.html for the discussion of the cases where the construction of the hyperbolas is possible.

See Also

Euler.html
EulerCircleProperty.html
FourEulerCircles.html
HyperbolaRectangular.html
NinePointsConic.html
OrthoRectangular.html
RectHypeCircumscribed.html
RectHyperbola.html
RectHyperbolasTangent4Lines.html
RectHypeThroughFourPts.html

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