5L-s-P1: 5L-Inscribed Conic Center

It is well known that in a system of 5 random lines a unique inscribed conic can be constructed. This conic is 5L-s-Co1 and 5L-s-P1 is the center of this conic.

In a 4-Line the Newton Line (QL-L1) is the locus of the centers of all 4L-inscribed conics. Consequently the Newton Lines of the 5 Component 4-Lines pass through the Center of the 5L-Inscribed Conic.

Construction

A simple way of construction of 5L-s-P1 is by drawing the Newton Lines (QL-L1) of two Component 4-Lines. The intersection point of these lines will be the center of the 5L-Inscribed Conic.

Coordinates

When using barycentric coordinates/coefficients:

L1=(1:0:0), L2=(0:1:0), L3=(0:0:1), L4=(l:m:n), L5=(L:M:N)

then 5L-s-P1 has coordinates:

( m n L (M – N) – M N l (m – n) :

n l M (N – L) – N L m (n – l) :

l m N (L – M) – L M n ( l – m) )

Properties

• 5L-s-P1 is collinear with 5G-s-P1, 5G-s-P2, 5G-s-P5 for every 5G-version of the 5L.
• 5L-s-P1 is the 4th harmonic point of 5G-s-P1 wrt 5G-s-P2 and 5G-s-P5 for every 5G-version of the 5L. See [66], QPG-message #1152.

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