QA-L-1: 5th Point Tangents
5th Point Tangents are the tangents of a conic through the vertices P1, P2, P3, P4 of the Quadrangle at a 5th point unequal to P1, P2, P3, P4.
Let P5 (u : v : w) be the 5th point.
This gives a very simple general formula for the coefficients of the tangent at this point.
p v w (q w - r v)
u (r2 v2 - q2 w2)
- The Crosspoint(P,Pi) wrt triangle Pj.Pk.Pl (i,j,k,l are different numbers from (1,2,3,4)) lies on the 5th point tangent of P. See Ref-13, keyword Crosspoint.
5th Point Tangent at QA-P1 (QA-Centroid)
p (q - r) / (2p + q + r)
(q2 - r2) (-p2 + q2 + r2)
5th Point Tangent at QA-P3 (Gergonne-Steiner Point)
p (b2(p+q)pr - c2(p+r)pq) / ((a2 - b2)pq + (a2 - c2)pr + a2qr - 2SA p2)
(q2 (a2 r2 - c2 p2) - r2 (a2 q2 - b2 p2)) *
(b2 r2 (p2 + q2 - r2)+ c2 q2 (p2 - q2 + r2) - 2 a2 q2 r2)
- This line passes through QA-P4 (Isogonal Center).