QL-2Co1: Pair of Inscribed Orthogonal Hyperbolas
Exactly 2 Inscribed Orthogonal Hyperbolas can be constructed in a Quadrilateral.
However only a Quadrilateral with obtuse angled Component Triangles have real Inscribed Orthogonal Hyperbolas. See Ref-52, Paris Pamfilos.
The intersections of QL-L1 and QL-Ci1 (not always real) are the centers of the Inscribed Orthogonal Hyperbolas of a Quadrilateral.
The intersections of the two inscribed orthogonal QL-hyperbolas give the orthocentric quadrangle of the in-/ex-centers of QL-Tr1.
All here mentioned properties are from Eckart Schmidt (see Ref-34, QFG-messages #1283-#1296), unless otherwise stated.
The axes can be constructed as follows:
- Let M1, M2 be the intersections of QL-Ci1 with the perpendicular bisector of the centers X, Y.
- Let C1 and C2 be circles round M1 and M2 through X and Y.
- Let Si,1 and Si,2 be the intersections of Mi.QL-P16 and the circle Ci.
- The lines Si,j.Sk,l (i ≠ k) are two pairs of orthogonal lines through the centers X, Y, which are the axes of the hyperbolas.
Dual properties
There are dual properties regarding Orthogonal Hyperbolas and Parabolas related to Quadrilaterals and Quadrangles.
•A random quadrilateral has only 1 Inscribed Parabola and 2 Inscribed Orthogonal Hyperbola’s.
•A random quadrangle has 2 Circumscribed Parabola’s and 1 Circumscribed Orthogonal Hyperbola.
These follow from properties of pencils and ranges of conics. All the conics touching the sides of a quadrilateral form a range. All the conics passing through the vertices of a quadrangle form a pencil.
A pencil of conics cuts a general line in pairs of points in involution. The double points of this involution show where just two conics in the pencil touch the line. Applying this to the line at infinity gives two parabolas in the pencil. Comparing this involution with the orthogonal involution at infinity, the pair of points common to both involutions lie on the rectangular hyperbola.
A range of conics has the dual properties.
The asymptotes are the tangents at infinity, obtained by joining the center to the two points at infinity on the hyperbola. If those two points are a pair in the orthogonal involution, then the asymptotes are perpendicular.
A pencil of conics includes one conic passing through a general point, but two conics touching a general line. A range of conics includes one conic touching a general line, but two conics passing through a general point. The unique conic in the range which touches the line at infinity is a parabola.
See Ref-34, Ken Pledger, QFG-messages #1285, #1294.
CT-equation Orthogonal Hyperbolas:
(-d + e + f)2 x2 + (d - e + f)2 y2 + (d + e - f)2 z2
- 2 (-d + e + f) (d - e + f) x y - 2 (-d + e + f) (d + e - f) x z - 2 (d + e - f) (d - e + f) y z
Ce1a/b = ( ( a2 - b2 + c2) (l m - l n - m n) (l m + l n - m n) + ( l m - l n + m n) W :
(-a2 + b2 + c2) (l m - l n - m n) (l m - l n + m n) + (-l m - l n + m n) W :
2 (c2 l2 m2 + c2 l2 n2 + c2 m2 n2 + 2 a2 l2 m n - 2 b2 l2 m n - 2 a2 l m2 n + 2 b2 l m2 n - 2 c2 l m n2) )
W = Sqrt [(a - b - c) (a + b - c) (a - b + c) (a + b + c) (m2 n2 + l2 m2 + l2 n2) +
2 l m n ((a2 + b2 + c2)2 (l + m + n) - 4 ((a4 + b2 c2) l + (b4 + a2 c2) m + (c4 + a2 b2) n))]
Properties:
- The intersections of QL-L1 and QL-Ci1 (not always real) are the centers of the Inscribed Orthogonal Hyperbolas of a Quadrilateral.
- The intersections of the two inscribed orthogonal QL-hyperbolas give the orthocentric quadrangle of the in-/ex-centers of QL-Tr1.
- The foci of QL-2Co1 a/b are the intersections of their axes and their QL-Tf1 image circle.
- The 8 contact points of the two inscribed orthogonal QL-hyperbolas lie on a conic.
- The QL-Diagonal Triangle is self-polar wrt the conic for the contact points of the inscribed orthogonal hyperbolas.