QL-Qu2: Kantor-Hervey Deltoid
QL-Qu2 is the only Deltoid that can be constructed tangent to the four lines of a Quadrilateral.
This Deltoid was used by Kantor and Hervey in order to prove their theorem that point QL-P3 was belonging to the 4 perpendicular bisectors of the Euler segments of the reference triangles.
It seems they are the oldest reference for this Deltoid. See reference at Ref-38.
More information about the subject also can be found in Bernard Keizer’s document at Ref-43 as well as at Ref-34, QFG #514.
Construction:
Another construction can be found at Ref-34, QFG messages #522 and #529:
- Let P be a point on QL-L2,
- Let line L be the perpendicular bisector of P.QL-P1,
- Let O = QL-Tf1(P),
- Let line T be parallel to L through the reflection of O in QL-P5.
- The lines T envelope QL-Qu2.
A construction of the axes of QL-Qu2 can be found at QFG messages #535 and #541.
Equation:
Equation QL-Qu2 in CT-notation (Bernard Gibert, August 15, 2013):
SIM4 (x + y + z)4
+ 4 SIM3 (x + y + z)3 MED B3L
+ 2 SIM2 (x + y + z)2 MED2 C3L
+ 4 SIM (x + y + z) MED3 K3L
+ MED4 H3 = 0
where:
SIM = c2 (l + m) (l - n) (m - n) - b2 (l - m) (m - n) (l + n) + a2 (l - m) (l - n) (m + n)
MED = (l - m) (m - n) (n - l)
B3L = b2 x - c2 x - a2 y + c2 y + a2 z - b2 z
C3L = -2 b2 c2 (3 x2 - 3 x y - 3 x z - 2 y z) + c4 (3 x2 - 6 x y + 3 y2 - z2) + b4 (3 x2 - y2 - 6 x z + 3 z2) + a4 (-x2 + 3 y2 - 6 y z + 3 z2) + a2 (2 c2 (3 x y - 3 y2 + 2 x z + 3 y z) + 2 b2 (2 x y + 3 x z + 3 y z - 3 z2))
K3L = b6 (x - z) (x - y - z) (x + y - z) - c6 (x - y) (x - y - z) (x - y + z) + a6 (y - z) (x + y - z) (x - y + z) - b4 c2 (x - y - z) (3 x2 - y2 - 3 x z - 7 y z) + b2 c4 (x - y - z) (3 x2 - 3 x y - 7 y z - z2) + a4 (c2 (x - y + z) (x2 - 3 y2 + 7 x z + 3 y z) - b2 (x + y - z) (x2 + 7 x y + 3 y z - 3 z2)) + a2 (-4 b2 c2 (x - y) (x - z) (y - z) + b4 (x + y - z) (7 x y + y2 + 3 x z - 3 z2) - c4 (x - y + z) (3 x y - 3 y2 + 7 x z + z2))
H3 = b8 (x - y - z)2 (x + y - z)2 + c8 (x - y - z)2 (x - y + z)2 + a8 (x + y - z)2 (x - y + z)2 - 4 b6 c2 (x - y - z)2 (x2 + x y - x z - 4 y z) - 4 b2 c6 (x - y - z)2 (x2 - x y + x z - 4 y z) + 2 b4 c4 (x - y - z)2 (3 x2 - y2 - 14 y z - z2) + a6 (-4 c2 (x - y + z)2 (x y + y2 - 4 x z - y z) + 4 b2 (x + y - z)2 (4 x y - x z + y z - z2)) + a4 (-2 b4 (x + y - z)2 (x2 + 14 x y + y2 - 3 z2) - 2 c4 (x - y + z)2 (x2 - 3 y2 + 14 x z + z2) + 4 b2 c2 (x4 + 7 x3 y - x2 y2 - 7 x y3 + 7 x3 z + 48 x2 y z + 7 x y2 z + 2 y3 z - x2 z2 + 7 x y z2 - 4 y2 z2 - 7 x z3 + 2 y z3)) + a2 (4 c6 (x - y + z)2 (x y - y2 + 4 x z - y z) + 4 b6 (x + y - z)2 (4 x y + x z - y z - z2) - 4 b4 c2 (7 x3 y + x2 y2 - 7 x y3 - y4 - 2 x3 z - 7 x2 y z - 48 x y2 z - 7 y3 z + 4 x2 z2 - 7 x y z2 + y2 z2 - 2 x z3 + 7 y z3) + 4 b2 c4 (2 x3 y - 4 x2 y2 + 2 x y3 - 7 x3 z + 7 x2 y z + 7 x y2 z - 7 y3 z - x2 z2 + 48 x y z2 - y2 z2 + 7 x z3 + 7 y z3 + z4))
Properties:
- QL-P3 is the Center of the Deltoid.
- QL-Ci4, the Hervey Circle is the inscribed circle of the Deltoid (note Bernard Keizer, 2013).
- QL-Qu2 is not only tangent to L1, L2, L3, L4 but also to the line parallel to the Newton Line QL-L1 through the Reflection of QL-P1 in QL-P5 (note Bernard Keizer, 2013), which is the asymptote of QL-Cu1 (see Ref-34, Bernard Keizer, QFG#2517).
- The 3 axes of QL-Qu2 (through QL-P3 and the 3 cusps) are parallel to the 3 asymptotes of Eckart’s Cubic QL-Cu2 (Eckart Schmidt, May 9, 2013).
- A congruent QL-Deltoid can be constructed with its cusps in the same direction with center QL-P4 and inscribed circle QL-Ci3. See Ref-34, QFG#514.