X901–X1000

Star-Named Centers, Ehrmann Points & Anticomplements

Star-named centers (Agena, Achernar, Algenib, Alcor), Ehrmann points, anticomplements, and midpoints of classical triangle centers.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.8571, 0.0000) R_a = Radius Point R_a (0.2857, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_901 = ISOGONAL CONJUGATE OF X(900) (1.0000, -0.0000) X_904 = -HIRST INVERSE OF X(256) (0.6163, 0.2816) X_906 = INTERSECTION OF LINES X(100)X(112) AND X(101)X(109) (-0.2109, 0.3287) X_907 = POINT ACRUX (0.1933, 0.5099) X_909 = POINT ADHAFERA (0.9286, -0.3499) X_911 = POINT ADHARA (0.8698, -0.4452) X_913 = POINT AGENA (0.8959, 0.3579) X_922 = POINT ALBALI (0.9307, 0.9656) X_935 = POINT ALCOR (0.8281, 0.4929) X_936 = INTERSECTION OF LINES X(1)X(2) AND X(3)X(9) (0.5138, 0.1768) X_944 = HOFSTADTER TRAPEZOID POINT (0.5306, 0.0708) X_952 = INTERSECTION OF X(1)X(5) AND X(3)X(8) (0.1267, 0.6191) X_962 = LONGUET-HIGGINS POINT (0.7551, 0.6582) X_970 = CENTER OF THE APOLLONIUS CIRCLE (0.2643, -0.2872) X_971 = INTERSECTION OF LINES X(3)X(9) AND X(4)X(7) (0.6441, 0.7770) X_992 = INTERSECTION OF LINES X(2)X(6) AND X(9)X(39) (0.2678, -0.0897) A B C X3 X1 X901 X904 X906 X907 X909 X911 X913 X922 X935 X936 X944 X952 X962 X970 X971 X992 (a, b, c) = (5, 6, 7) | AB = 1

X901–X1000 (Star-Named Centers, Ehrmann Points & Anticomplements) — 16 representative centers (from 100 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X_903 — X_1-BLAIKIE TRANSFORM OF X_2

X903-BLAIKIE TRANSFORM OF X2

James Blaikie (1847-1929) proposed the following problem. Let O be any point in the plane of triangle ABC, and let any straight line g through O meet BC in P, CA in Q, AB in R; then, if points P', Q', R' be taken on the line so that PO = OP', QO = OQ', RO = OR'. Prove that AP', BQ', CR' concur.
X_933 — X_4-CROSS CONJUGATE OF X_250

X933-CROSS CONJUGATE OF X250

Let P be a point on the circumcircle, and let

(Oab) = circle through A tangent to line BC at B

(Oac) = circle through A tangent to line BC at C

Ba = point other than A where the line PB meets (Oab)

Ca = point other than A where the line PC meets (Oac)

Hab = orthocenter of APBa

Hac = orthocenter of APCa

X_935 — POINT ALCOR

X935POINT ALCOR

X935 = trilinear pole of line X6X67 (the radical axis of the orthocentroidal circle and Dao-Moses-Telv circle)
X_953 — ISOGONAL CONJUGATE OF X_952

X953ISOGONAL CONJUGATE OF X952

For a construction of X953 involving the Euler line and X110, see Antreas Hatzipolakis and César Lozada, Hyacinthos 26050.
X_970 — CENTER OF THE APOLLONIUS CIRCLE

X970CENTER OF THE APOLLONIUS CIRCLE

The Apollonius circle is described at X181 as the circle tangent to the three excircles and encompassing them. That X970 is its center was noted on New Year’s Day, 2002, by Paul Yiu. (Hyacinthos #4619-4623)