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At this site you will find mathematical subjects I investigated. All pages are in English. Some pages are also available in Dutch. When a change of language is possible it is shown at the top of the page.
Chris van Tienhoven, the Netherlands
Op deze site vindt u enige wiskundige onderwerpen waar ik onderzoek naar gedaan heb. Alle pagina's zijn in het Engels. Sommige pagina's zijn ook in het Nederlands. Wanneer er een taalkeuze is, dan staat dit bovenaan de pagina d.m.v. een (NL-)vlag.
Chris van Tienhoven, Nederland
QL-Tf11: QL-QuadriPolar

The QL-QuadriPolar is an equivalent of the Tripolar (also named Trilinear Polar) in a trilateral. It transforms in a Quadrilateral “harmonically” a point into a line.
QL-Tf11(P) = QL-Tf2(DT-TP(P)), where DT-TP = Trilinear Polar wrt the QL-Diagonal Triangle QL-DT (=QL-Tr1).
QL Tf11 QuadriPolar 01
The combination QL-Tf10/QL-Tf11 in a Quadrilateral is the equivalent of the combination QA-Tf10/QA-Tf11 in a Quadrangle.
In particular QL-Tf11 is the dual of QA-Tf10 and has the same coordinates as QA-Tf10 when substituting (p:q:r) >(l:m:n).
QL-Tf11 is used by the 1st construction of the Involutary Centerline QL-Tf8.
Let P = (x:y:z), then QL-Tf11(P)=
(l (2 l x + m y + n z) : m (l x + 2 m y + n z) : n (l x + m y + 2 n z))
Let P = (x:y:z), then QL-Tf11(P)=
(l2 x : m2 y : n2 z)

QL-Tf11(QL-Tf10(L)) = L and QL-Tf10(QL-Tf11(P)) = P.
QL-Tf11(P) also can be obtained as DT-TP(QA-Tf2*(P)), where QA-Tf2* = QA-Tf2-transformation wrt the dual QA with vertices Pi=DT-TP(Li) (i=1,2,3,4). Therefore it is also QA-Tf11(P) wrt the dual QA. See QL-8. 


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