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CU-4P-cHe1 CU-inscribed Complete Hexagon

Definitions

Definition 1: A hexagon, or 6-gon, is a geometrical figure consisting of 6 consecutive vertices, with no three of them being collinear.

Definition 2: A complete hexagon is a geometric figure formed by 6 consecutive vertices, with no three of them being collinear. In addition, it includes the points of intersection of the 3 pairs of opposite sides.

In other words, a complete hexagon comprises 6 consecutive vertices and the points of intersection of the 3 pairs of opposite sides, forming a total of 9 distinct points.

Theorem: A complete hexagon can be inscribed in a cubic curve using only four initial points.

Infovisual CU-4P-cHe1-infovisual-cvt-01.png
Construction

The procedure to construct the CU-inscribed Hexagon (6-gon) using 4 random points on CU is:

  1. Start with 4 random points P1,P2,P3,P4 on a reference cubic CU.
  2. Draw line P1P2 intersecting CU again in S12.
  3. Draw line P2P3 intersecting CU again in S23,
  4. Draw line P3P4 intersecting CU again in S34,
  5. Draw line P4S12 intersecting CU again in P5,
  6. Draw line P5S23 intersecting CU again in P6.
  7. Draw line P6S34 intersecting CU again in P1 !

Last step easily can be proven using the CU Point Addition Method CU-4:

CU Point Validation
  1. Use the property that for 3 collinear points P, Q, and R, the sum P + Q + R = N.
  2. S12 = N – P1 – P2
  3. S23 = N – P2 – P3
  4. S34 = N – P3 – P4
  5. P5 = N – P4 – S12 = N – P4 – (N – P1 – P2) = P1 + P2 – P4
  6. P6 = N – P5 – S23 = N – P5 – (N – P2 – P3) = P2 + P3 – P5
  7. Px = N – P6 – S34 = N – (P2 + P3 – P5) – (N – P3 – P4) = -P2 + P4 + P5 = -P2 + P4 + (P1 +P2 – P4) = P1
Connection with Eckart’s Hexagon Theorem

Eckart’s Hexagon Theorem (QPG#1799) states that given 3 collinear points P,Q,R and a starting point X1, then a closed Hexagon can be constructed. The 6 vertices will be coconic.

The way of construction of the Hexagon is similar to the construction of the CU-4P-Hexagon, only the starting points are different, there are 3 collinear points and 1 other point.

Applications
  • The 3 Quasi-Miquel Triangles (CU-IP-3P1) form together a CU-inscribed Complete Hexagon.
  • The 9P-SumPoints and surrounding points form a CU-inscribed Complete Hexagon. See CU-9P-P1.


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