QL-Tr1: QL-Diagonal Triangle


The Diagonal Triangle of the Quadrilateral (L1, L2, L3, L4) is the Triangle bounded by lines S12.S34, S13.S24, S14.S23, where Sij = intersection point Li ^ Lj, where i and j are different numbers and ∈ (1,2,3,4).
The connection of the intersection points of 2 adjacent lines and the intersection point of the remaining 2 lines produces a side of the QL-Diagonal Triangle. By doing this for all 3 possible combinations of 2 sets of 2 adjacent lines the QL-Diagonal Triangle is produced.
 
QL-Tr1-DiagonalTriangle-00
 
Area/Radius:  
Area QL-Diagonal Triangle in CT-notation:
                                                                                     4 l2 m2 n2 Δ                                    .
(l m - l n - m n) (l m + l n - m n) (l m - l n + m n)

Radius2 Circumcircle QL-Diagonal Triangle in CT-notation:
l2 m2 n2 
* (a2 l2 - a2 l m - b2 l m + c2 l m + b2 m2) (a2 l2 - a2 l n + b2 l n - c2 l n + c2 n2) (b2 m2 + a2 m n - b2 m n - c2 m n + c2 n2)
/ (4 Δ2 (l m - l n - m n)2 (l m + l n - m n)2 (l m - l n + m n)2 )
 
Area QL-Diagonal Triangle in DT-notation:
            S /2
Radius2 Circumcircle QL-Diagonal Triangle in CT-notation:
            a2 b2 c2 /(4 S2)

Properties:
  • The QL-Diagonal Triangle is the AntiCevian Triangle of each of the 4 Component Triangles Li.Lj.Lk, where Ll is the perspectrix, with (i,j,k,l) (1,2,3,4).
  • QL-Tr1 is self-polar wrt every conic inscribed in the Reference Quadrilateral.