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X₄₈₅–X₅₁₀¶

Vecten Square Centers & Isoscelizer Centers

Inner and outer Vecten points and square-associated centers (Vierkanten in een driehoek, X₄₈₅–X₄₉₅), harmonic conjugates, and isoscelizer-associated centers (X₅₀₃–X₅₁₀).

X₄₈₅–X₅₁₀ (Vecten Square Centers & Isoscelizer Centers) — 13 representative centers (from 26 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

Group Preamble — Centers X₄₈₅-X₄₉₅ (triangle centers associated with squares)

Centers X₄₈₅-X₄₉₅, and also X₃₇₁ and X₃₇₂, are Vierkanen in een driehoek: triangle centers associated with squares.

Group Preamble — Centers X₅₀₃-X₅₁₀ (Isoscelizer points: X₅₀₃-X₅₁₀)

Centers X₅₀₃-X₅₁₀, and also centers X₁₇₃, X₁₇₄, X₂₅₈, and X₃₅₁-X₃₆₄ are associated with isoscelizers.

A line L_A perpendicular to the internal bisector line of A is an A-isoscelizer.. Suppose X is a point not on a sideline of ABC, and let

L(A,X) = the A-isoscelizer passing through X; E(A,X) = L(A,X)∩AC; \(F(A,X) =L(A\),X)∩AB; T(A,X) = the triangle wth vertices A, E(A,X), F(A,X); \(H(A,X) =A-altitude of T(A\),X); D(A,X) = distance between E(A,X) and F(A,X); X(A) = distance between E(A,X) and F(A,X);

Cyclically define L(B,X), E(B,X), … , X(B) and L(C,X), E(C,X), … , X(C).

Each center, X₅₀₃ to X₅₁₀, is defined by Peter Yff as the point X of concurrence of isoscelers satisfying certain conditions.

Geometer’s Sketchpad sketches for centers X₅₀₃-X₅₁₀ were contributed by Peter Moses, May 7, 2005.

X₄₈₅VECTEN POINT

X₄₈₅ = complement of X₄₈₈

X₄₈₆INNER VECTEN POINT

X₄₈₆ = complement of X₄₈₇

X₄₈₇ANTICOMPLEMENT OF X₄₈₆

X₄₈₇ is the anticomplement of X₄₈₆.

X₄₈₈ANTICOMPLEMENT OF X₄₈₅

X₄₈₈ is the anticomplement of X₄₈₅.

X₄₈₉CEVAPOINT OF X₂₀ AND X₄₈₇

X₄₈₉ is the cevapoint of X₂₀ and X₄₈₇.

X₄₉₀CEVAPOINT OF X₂₀ AND X₄₈₈

X₄₉₀ is the cevapoint of X₂₀ and X₄₈₈.

X₄₉₁CEVAPOINT OF X₂ AND X₄₈₇

X₄₉₁ is the cevapoint of X₂ and X₄₈₇.

X₄₉₂CEVAPOINT OF X₂ AND X₄₈₈

X₄₉₂ is the cevapoint of X₂ and X₄₈₈.

X_493 — 1st VAN LAMOEN HOMOTHETIC CENTER

X₄₉₃1st VAN LAMOEN HOMOTHETIC CENTER

X₄₉₃ is a homothetic center of triangles associated with squares that circumscribe ABC. For details, see Vierkanten in een driehoek: 4. Ingeschreven vierkanten (Floor van Lamoen, 4/27/98)

X_494 — 2nd VAN LAMOEN HOMOTHETIC CENTER

X₄₉₄2nd VAN LAMOEN HOMOTHETIC CENTER

X₄₉₄ is the homothetic center of triangle ABC and the Lucas(-1) homothetic triangle; see X₃₇₁.

X₄₉₅JOHNSON MIDPOINT

X₄₉₅ is the midpoint of segments C₁-to-P₁, C₂-to-P₂, C₃-to-P₃ in the Johnson four-circle configuration.

X_496 — {X_1,X_5}-HARMONIC CONJUGATE OF X_495

X₄₉₆{X₁,X₅}-HARMONIC CONJUGATE OF X₄₉₅

X₄₉₆ is the {X₁,X₅}-harmonic conjugate of X₄₉₅.

X_497 — CROSSPOINT OF GERGONNE POINT AND NAGEL POINT

X₄₉₇CROSSPOINT OF GERGONNE POINT AND NAGEL POINT

X₄₉₇ is the crosspoint of Gergonne point and Nagel point.

X_498 — YFF CONCURRENT CONGRUENT CIRCLES POINT

X₄₉₈YFF CONCURRENT CONGRUENT CIRCLES POINT

X₄₉₈ and X₄₉₉ are harmonic conjugate points with respect to X₁ and X₂, in analogy with such pairs with respect to X₁, X₄ and with respect to X₁, X₅.

X_499 — {X_1,X_2}-HARMONIC CONJUGATE OF X_498

X₄₉₉{X₁,X₂}-HARMONIC CONJUGATE OF X₄₉₈

X₄₉₉ is the harmonic conjugate of X₄₉₈ with respect to X₁ and X₂.

X_500 — ORTHOCENTER OF THE INCENTRAL TRIANGLE

X₅₀₀ORTHOCENTER OF THE INCENTRAL TRIANGLE

X₅₀₀ is the orthocenter of the incentral triangle.

X₅₀₁MIQUEL ASSOCIATE OF INCENTER

Miquel’s theorem states that if A', B', C' are points (other than A, B, C) on sidelines BC, CA, AB, respectively, then the circles AB'C', BC'A', CA'B' meet at a point. Suppose P is a point and A' = P∩BC, B' = P∩CA, C' = P∩AB; the point in which the three circles is the Miquel associate of P. (Paul Yiu, 7/6/99)