RectHypeCircumscribed

1. Rectangular Hyperbolas Circumscribed on a Triangle

From the discussion in OrthoRectangular.html we know that each rectangular hyperbola circumscribed on a triangle has its center on the Euler circle. Following the analysis in the previous reference we can determine the hyperbola from the location of its center C. The figure below demonstrates this and gives a means to see the various hyperbolas by moving C on the Euler circle (press CTRL+2 to switch to the move-on-contour tool).

(1) From the discussion in the previous cited reference and the one in EulerCircleProperty.html we know that each point C determines an angle(CBA) and the direction of one asymptotic line coincides with the bisector A₁J of angle(A₂A₁E), E being the symmetric of A₂ w.r. to C. It is angle(AA₁J) = (1/4)angle(ABC). Hence the asymptotics of the hyperbola can be found from the location of C.
(2) To determine the hyperbola take the projections D, F of E on the asymptotics and determine the vertex G of the hyperbola, lying on the bisector of angle(DCF) and having CG² = 2*ED*EF. Having C and G, draw the hyperbola using the [Rectangular Hyperbola] tool.

The small diagram shows (part of) the periodic graph of the function of length of y=|CG| (axis of the hyperbola) as a function of the polar angle of the asymptotic line. The lower peaks correspond to y=0, and are taken when C is at the feet of the altitudes, where the hyperbolas degenerate to two orthogonal lines. The upper local maxima correspond to the positions of C, for which the fourth intersection point I of the circumcircle, becomes identical with a vertex of the triangle. Then the hyperbola is tangent at this vertex to the circumcircle.

2. Other construction

Next figure illustrates another method to construct the rectangular hyperbola circumscribing triangle ABC and having its center at the point I of the Euler circle of the triangle.
Take the symmetrics {D,G,...} of the vertices of the triangle with respect to I and draw the general conic passing through five out of the six created points.
The figure shows also the locus of perspectors of the circumscribed rectangular hyperbolas which is the orthic axis of the triangle.
Triangle EFG is a precevian triangle of ABC, whose sides pass through the vertices of ABC and coincide there with the tangents of the circumscribed hyperbola.

Remark When I obtains the position of the middle of a side of ABC the corresponding rectangular hyperbola is the Antiparallel Hyperbola of the triangle with respect to this side, discussed in AntiparallelHyperbola.html .

1. Rectangular Hyperbolas Circumscribed on a Triangle

2. Other construction

See Also