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.

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.

EulerCircleProperty.html

FourEulerCircles.html

HyperbolaRectangular.html

NinePointsConic.html

OrthoRectangular.html

RectHypeCircumscribed.html

RectHyperbola.html

RectHyperbolasTangent4Lines.html

RectHypeThroughFourPts.html

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