QA-P33: Centroid of the Orthocenter Quadrangle

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QA-P33 is the QA-Centroid of the quadrangle formed by the Orthocenters of the Component Triangles of the Reference Quadrangle.

                       

 

Coordinates: 

1^st CT-Coordinate:

+a⁴ q² r² – S_B c² p² q² – S_C b² p² r² + p q r (3 S_B S_C (p + q + r) + S² p)

 

1^st DT-Coordinate:

            S² p⁴ + 2 a⁴ q² r² - (S² + 2 S_B c²) p² q² - (S² + 2 S_C b²) p² r²

Properties:

• QA-P33 lies on these QA-lines:

–        QA-P1.QA-P32          (3 : -2)

–        QA-P12.QA-P14        (3 : -1 => QA-P33 =Complem. of QA-P12 wrt QA-Tr3)

–        QA-P24.QA-P37        (1 :  1 => QA-P33 = Midpoint QA-P24.QA-P37)

• QA-P1.QA-P32.QA-P33 // QA-P2.QA-P4.QA-P6 = QA-L2.
• QL-P2 is the Centroid of the QL-Triangle formed by the 3 QL-versions of QA-P33 (note Eckart Schmidt).
• The QA-Orthopole (QA-Tf3) of QA-P33 is a point on the line QA-P23.QA-P33 (ratio -1:3).
• The area of the Orthocenter Quadrangle equals the area of the Reference Quadrangle.

 

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