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X₁₁₃–X₁₃₉¶

Centers on the Nine-Point Circle & Orthic Centers

Points on the nine-point circle (X₁₁₃–X₁₃₉), centers of the Kiepert, Feuerbach, and Jerabek rectangular hyperbolas, and centers transferred to the orthic triangle.

X₁₁₃–X₁₃₉ (Centers on the Nine-Point Circle & Orthic Centers) — 14 representative centers (from 27 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

Group Preamble — Centers X₁₁₃-X₁₃₉ (Points on the nine-point circle)

Centers X₁₁₃-X₁₃₉ lie on the nine-point circle.

Suppose that X is a point on the nine-point circle, and let X' be the reflection of X in the orthocenter, H. Then X is the anticenter of the cyclic quadrilateral ABCX’. Let H_A be the orthocenter of triangle BCX, Let H_B be the orthocenter of CAX, and let H_C be the orthocenter of triangle ABX. Then the quadrilateral \(HH_{A}\)H_BH_C is homothetic to and congruent to the cyclic quadrilateral ABCX’, and X is the center of homothety. (Randy Hutson, 9/23/2011)

X_113 — JERABEK ANTIPODE

X₁₁₃JERABEK ANTIPODE

X₁₁₃ = complementary conjugate of X₃₀

X_114 — KIEPERT ANTIPODE

X₁₁₄KIEPERT ANTIPODE

X₁₁₄ is the QA-P30 center (Reflection of QA-P₂ in QA-P11) of quadrangle ABCX₂ (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/58-qa-p30.html)

X_115 — CENTER OF KIEPERT HYPERBOLA

X₁₁₅CENTER OF KIEPERT HYPERBOLA

X₁₁₅ is the QA-P₂ center (Euler-Poncelet Point) of quadrangle ABCX₂ (see http://www.chrisvantienhoven.nl/quadrangle-objects/10-mathematics/quadrangle-objects/12-qa-p2.html)

X_116 — MIDPOINT OF X_4 AND X_103

X₁₁₆MIDPOINT OF X₄ AND X₁₀₃

X₁₁₆ is the midpoint of X₄ and X₁₀₃.

X_117 — MIDPOINT OF X_4 AND X_109

X₁₁₇MIDPOINT OF X₄ AND X₁₀₉

X₁₁₇ is the midpoint of X₄ and X₁₀₉.

X_118 — MIDPOINT OF X_4 AND X_101

X₁₁₈MIDPOINT OF X₄ AND X₁₀₁

X₁₁₈ is the midpoint of X₄ and X₁₀₁.

X_119 — FEUERBACH ANTIPODE

X₁₁₉FEUERBACH ANTIPODE

X₁₁₉ = complement of X₁₀₄

X_120 — X_105-OF-MEDIAL-TRIANGLE

X₁₂₀-OF-MEDIAL-TRIANGLE

X₁₂₀ = X₁₀₅-of-medial triangle.

X_121 — X_106-OF-MEDIAL-TRIANGLE

X₁₂₁-OF-MEDIAL-TRIANGLE

X₁₂₁ = X₁₀₆-of-medial triangle.

X_122 — X_107-OF-MEDIAL-TRIANGLE

X₁₂₂-OF-MEDIAL-TRIANGLE

X₁₂₂ = reflection of X₁₃₃ in X₅

X_123 — X_108-OF-MEDIAL-TRIANGLE

X₁₂₃-OF-MEDIAL-TRIANGLE

X₁₂₃ = complement of X₁₀₈

X_124 — X_109-OF-MEDIAL-TRIANGLE

X₁₂₄-OF-MEDIAL-TRIANGLE

X₁₂₄ = X₁₀₉-of-medial triangle

X_125 — CENTER OF JERABEK HYPERBOLA

X₁₂₅CENTER OF JERABEK HYPERBOLA

X₁₂₅ = complement of X₁₁₀

X_126 — X_111-OF-MEDIAL-TRIANGLE

X₁₂₆-OF-MEDIAL-TRIANGLE

X₁₂₆ = X₁₁₁-of-medial triangle.

X_127 — X_112-OF-MEDIAL-TRIANGLE

X₁₂₇-OF-MEDIAL-TRIANGLE

X₁₂₇ = reflection of X₁₃₂ in X₅

X_128 — X_74-OF-ORTHIC-TRIANGLE

X₁₂₈-OF-ORTHIC-TRIANGLE

X₁₂₈ = X₇₄-of-orthic triangle.

X_129 — X_98-OF-ORTHIC-TRIANGLE

X₁₂₉-OF-ORTHIC-TRIANGLE

X₁₂₉ = X₉₈-of-orthic triangle.

X_130 — X_99-OF-ORTHIC-TRIANGLE

X₁₃₀-OF-ORTHIC-TRIANGLE

X(130 lies on the nine-point circle

X_131 — INTERSECTION OF LINES X_3X_125 AND X_4X_135

X₁₃₁INTERSECTION OF LINES X₃X_125 AND X₄X_135

X₁₃₁ = X₁₀₂-of-orthic triangle if ABC is acute.

X_132 — X_2X_107∩X_4X_32

X₁₃₂∩X₄X_32

X₁₃₂ = midpoint of X₄ and X₁₁₂

X_133 — INTERSECTION OF LINES X_4X_74 AND X_5X_122

X₁₃₃INTERSECTION OF LINES X₄X_74 AND X₅X_122

X₁₃₃ = X₁₀₆-of-orthic triangle is ABC is acute.

X_134 — X_107-OF-ORTHIC-TRIANGLE

X₁₃₄-OF-ORTHIC-TRIANGLE

X₁₃₄ = X₁₀₇-of-orthic triangle

X_135 — INTERSECTION OF LINE X_4X_131 AND X_25X_114

X₁₃₅INTERSECTION OF LINE X₄X_131 AND X₂₅X_114

X₁₃₅ = X₁₀₈-of-orthic-triangle if ABC is acute

X_136 — INTERSECTION OF LINE X_4X_110 AND X_25X_132

X₁₃₆INTERSECTION OF LINE X₄X_110 AND X₂₅X_132

X₁₃₆ =X₁₀₉-of-orthic triangle if ABC is acute

X_137 — X_110-OF-ORTHIC-TRIANGLE

X₁₃₇-OF-ORTHIC-TRIANGLE

X₁₃₇ = X₁₁₀-of-orthic triangle

X_138 — X_111-OF-ORTHIC-TRIANGLE

X₁₃₈-OF-ORTHIC-TRIANGLE

X₁₃₈ = X₁₁₁-of-orthic triangle

X_139 — X_112-OF-ORTHIC-TRIANGLE

X₁₃₉-OF-ORTHIC-TRIANGLE

X₁₃₉ = X₁₁₂-of-orthic-triangle