8L-s-L1: 8L-Keizer-Schmidt Line
The Keizer-Schmidt Line 8L-s-L1 in an 8-Line (Octalateral) is the equivalent of the 8P-Cayley Bacharach Point 8P-s-P1 in an 8-Point (Octangle).
It was developed by Eckart Schmidt and Bernard Keizer during an extensive discussion at the Quadri-Figures-Group. See Ref-34, search for keyword “ninth” to find the involved messages and especially messages #2511, #2513, #2515-#2519, #2561, #2640, #2641, #2644, #2662, #2663, #2677, #2678, #2679, #2691, #2693.
It is defined as follows: Let L1,...,L8 be eight distinct lines in the plane, no four of which concurrent, and no seven of them tangent to a conic. There exists a unique ninth line L9 such that every curve of class 3 tangent to L1, …..,L8, also will be tangent to L9.
Degree vs. Class of an algebraic curve
Note that in the Cayley-Bacharach Theorem for 8 points delivering a ninth point ,these points lie on a curve of degree 3. However in the equivalent for 8 lines delivering a ninth line, these points are tangent to a curve of class 3.
The degree of an algebraic curve Cv may be defined as the number of intersection points that can be found when intersecting Cv with a line (counting multiplicities and including imaginary points).
The class of an algebraic curve Cv may be defined as the number of tangents that may be drawn to Cv from a point not on Cv (counting multiplicities and including imaginary tangents).
The degree of an algebraic curve Cv may be defined as the number of intersection points that can be found when intersecting Cv with a line (counting multiplicities and including imaginary points).
The class of an algebraic curve Cv may be defined as the number of tangents that may be drawn to Cv from a point not on Cv (counting multiplicities and including imaginary tangents).
Construction
8L-s-L1 is proposed and constructed by Eckart Schmidt as follows:
Consider eight lines and take four for a reference QL and its dual QA,
... duality gives eight points for the eight lines,
... take their Cayley-Bacharach ninth point,
... its dual will give the Cayley-Bacharach ninth line.
The construction is independent of the chosen reference QL.
See Ref-34, QFG#2511, QFG#2662.
The mentioned QL-duals are line-point-dual QL-Tf10 and point-line-dual QL-Tf11.
The mentioned Cayley-Bacharach ninth point is 8P-s-P1.
Consider eight lines and take four for a reference QL and its dual QA,
... duality gives eight points for the eight lines,
... take their Cayley-Bacharach ninth point,
... its dual will give the Cayley-Bacharach ninth line.
The construction is independent of the chosen reference QL.
See Ref-34, QFG#2511, QFG#2662.
The mentioned QL-duals are line-point-dual QL-Tf10 and point-line-dual QL-Tf11.
The mentioned Cayley-Bacharach ninth point is 8P-s-P1.
Example
The eight angle bisectors of a Quadrigon (4-Gon) have a Cayley-Bacharach ninth line 8L-s-L1.
In this case the curve of class 3 is not known, but when we consider mentioned 8 lines to be tangent to some curve of class 3, then the ninth line 8L-s-L1 always will be tangent too.
Special properties for this configuration:
• The QL-Tf2-image of QG-P17.QG-P18 is the ninth CB-line wrt the eight angle bisectors of a Quadrigon.
• The 3 CB-lines of the Component Quadrigons of a Quadrilateral have a common point.
• This common point has 1st DT-coordinate: m2 n2 SA2(l2 SB SC a2 + m2 SC S2 + n2 SB S2).
See Ref-34, Eckart Schmidt, QFG#2677.
• The 3 CB-lines of the Component Quadrigons of a Quadrilateral have a common point.
• This common point has 1st DT-coordinate: m2 n2 SA2(l2 SB SC a2 + m2 SC S2 + n2 SB S2).
See Ref-34, Eckart Schmidt, QFG#2677.