X3001–X3118

Central Line Intersections, Stevanović Points & T/G Mappings

Pairwise intersections of the Brocard, de Longchamps, Gergonne, Lemoine, Nagel, orthic, Soddy, and Fermat axes (X3001–X3019), the 4th–10th Stevanović points (X3020–X3026), antipodal circle mappings and complements (X3027–X3111), and Quang Tuan Bui’s TgX and GtX mappings (X3112–X3118).
< 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_3002 = (BROCARD AXIS)∩(GERGONNE LINE) (1.1771, 1.2203) X_3003 = (BROCARD AXIS)∩(ORTHIC AXIS) (0.8684, 0.7105) X_3008 = (GERGONNE LINE)∩(NAGEL LINE) (1.4558, 1.0998) X_3010 = (LEMOINE AXIS)∩(SODDY LINE) (1.0519, 1.3447) X_3011 = (NAGEL LINE)∩(ORTHIC AXIS) (1.0051, 0.6582) X_3017 = (NAGEL LINE)∩(FERMAT AXIS) (0.6027, 0.2639) X_3018 = (ORTHIC AXIS)∩(FERMAT AXIS) (-0.0536, 1.0628) X_3020 = 4th STEVANOVIC POINT (0.7047, 0.0418) X_3021 = 5th STEVANOVIC POINT (0.3559, 0.3226) X_3028 = INCIRCLE-ANTIPODE OF X(3024) (0.4579, 0.4371) X_3031 = INTERSECTION X(10)X(125)∩X(74)X(573) (-0.6264, 1.3112) X_3044 = INTERSECTION X(54)X(114)∩X(99)X(184) (0.8676, -0.0249) X_3046 = INTERSECTION X(54)X(118)∩X(101)X(184) (0.8038, -0.0821) X_3052 = INTERSECTION X(31)X(42)∩X(32)X(220) (0.6825, 0.3888) X_3053 = INTERSECTION X(3)X(6)∩X(4)X(230) (0.7332, 0.4873) X_3054 = CENTER OF EVANS CONIC (0.4976, 0.1571) A B C X3 X1 X3002 X3003 X3008 X3010 X3011 X3017 X3018 X3020 X3021 X3028 X3031 X3044 X3046 X3052 X3053 X3054 (a, b, c) = (5, 6, 7) | AB = 1

X3001–X3118 (Central Line Intersections, Stevanović Points & T/G Mappings) — 16 representative centers (from 118 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Antipodal Pairs on Circles

In response to Stevanovic’s findings (X3000 to X3006), Peter Moses noted (Hyacinthos, 12/9/2004) a method for mapping a pair of antipodal points on one circle to an antipodal pair on another circle. The method depends on centers of similitude:

Suppose O1 and O2 are circles, that P is on O1 and that P' is the antipode of P on O1. Let U be the internal center of similitude (insimilicenter) of O1 and O2, and V the exsimilicenter. Define Q = PU ∩P' V and Q' = PV ∩P' U. Then on O2, point Q' is the antipode of Q. Moreover, the lines PP' and QQ' are parallel.

The method follows from introductory material on centers of similitude (e.g., H. S. M. Coxeter, Introduction to Geometry, 2nd edition, page 70, which cites Nathan Altshiller-Court, College Geometry, 2nd edition, page 184). If you have The Geometer’s Sketchpad, you can view Antipodal Pairs (AntipodalPairs.gsp).

In the following examples, suppose P =\(p : q : r\) (trilinears).

Example 1. O1 = circumcircle and O2 = incircle. In this case, insimilicenter(O1, O2) = X55, and exsimilicenter(O1, O2) = X56. The antipodal points on O2 given by the Moses mapping are

Q = ((b - c)2p + a(bq + cr))(b + c - a) : :

Q' = ((b + c)2p + a(bq + cr))/(b + c - a) : : .

Example 2. O1 = circumcircle and O2 = Apollonius circle, with center X970. Here, insimilicenter(O1, O2) = X573, and exsimilicenter(O1, O2) = X386. A point on O2 is

Q' = (a + b)(a + c)(b + c)2p - a(bc + ca + ab + \(b^{2}\) + \(c^{2})\)(bq + cr) : : .

Example 3. O1 = circumcircle and O2 = Spieker circle. In this case, insimilicenter(O1, O2) = X958, and exsimilicenter(O1, O2) = X1376.

Q = (a - b - c)[((b + c)(b - c)2 + \(a(b^{2}\) + \(c^{2})\))p + (ab + ac + 2bc + \(a^{2})\)(bq + cr)] : :

Q' = (\(b^{3}\) + \(c^{3}\) - \(ab^{2}\) - \(ac^{2}\) - \(bc^{2}\) - \(b^{2}\)c)p + (ab + ac - 2bc - \(a^{2})\)(bq + cr) : :

Example 4. O1 = incircle and O2 = Spieker circle. In this case, insimilicenter(O1, O2) = X2, and exsimilicenter(O1, O2) = X8.

Q = (bp + cq)/a : (cr + ap)/b : (ap + bq)/c = complement of P

Q' = 2abc(b + c)p - bc(a - b - c)(bq + cr) : :

Example 5. O1 = circumcircle and O2 = sine-triple-angle circle. In this case, insimilicenter(O1, O2) = X1147, and exsimilicenter(O1, O2) = X184. A point on O2 is

Q = a[\(a(a^{2}\) - \(b^{2})\)(\(a^{2}\) - \(c^{2})\)p - \(a^{2}(b^{2} -a^{2} + c^{2})\)(bq + cr)] : :

Group Preamble — T and G Mappings (Centers X3112-X3118)

This section presents points X3112 to X3118 as representative of points introduced by Quang Tuan Bui in Hyacinthos, August 5, 2006.

Suppose X is a point not on a sideline of ABC. Let

gX = isogonal conjugate of X
tX = isotomic conjugate of X
tgX = isotomic conjugate of gX
gtX = isogonal conjugate of tX
Gt = intersection of lines X(tX) and (gX)(gtX)
Tg = intersection of lines X(gX) and (tX)(tgX)

Then the points A, B, C, gX, tX, Tg, Gt are on a conic. As a circumconic, it is the image under the isogonal conjugate mapping of line X(gtX). It is also the image under the isotomic conjugate mapping of line X(tgX).

If X =\(x : y : z\) (trilinears), then

Gt = (\(b^{2}\) - \(c^{2})\)/(\(x(b^{2}\)\(y^{2}\) - \(c^{2}\)\(z^{2})\)) : (\(c^{2}\) - \(a^{2})\)/(\(y(c^{2}\)\(z^{2}\) - \(a^{2}\)\(x^{2})\)) : (\(a^{2}\) - \(b^{2})\)/(\(z(a^{2}\)\(x^{2}\) - \(b^{2}\)\(y^{2})\))

Tg = (\(b^{2}\) - \(c^{2})\)/(\(x(y^{2}\) - \(z^{2})\)) : (\(c^{2}\) - \(a^{2})\)/(\(y(z^{2}\) - \(x^{2})\)) : (\(a^{2}\) - \(b^{2})\)/(\(z(x^{2}\) - \(y^{2})\))

The X31-conic passes through Xi for I = 75, 92, 313, 321, 561, 1441, 1821, 1934, 2995, 2997.
The X32-conic passes through Xi for I = 76, 264, 276, 290, 300, 301, 308, 313, 327, 349, 1502, 2367.
The X76-conic passes through Xi for I = 6, 32, 83, 213, 729, 981, 1918, 1974, 2207, 2281, 2422.
X_3003 — (BROCARD AXIS)∩(ORTHIC AXIS)

X3003(BROCARD AXIS)∩(ORTHIC AXIS)

lies on the cubics K277, K489, K1170, K1171, and these lines: on lines {2, 3260}, {3, 6}, {30, 6128}, {31, 8606}, {37, 7359}, {44, 13006}, {53, 235}, {54, 43753}, {74, 32681}, {111, 9060}, {112, 32710}, {114, 45921}, {115, 11799}, {157, 1974}, {160, 6467}, {186, 8749}, {206, 40947}, {230, 231}, {237, 2393}, {248, 1177}, {393, 847}, {395, 40696}, {396, 40695}, {419, 1632}, {524, 34990}, {607, 2178}, {608, 2164}, {615, 8963}, {648, 44375}, {672, 22059}, {895, 35298}, {1084, 23967}, {1100, 2649}, {1194, 5306}, {1196, 34811}, {1213, 16699}, {1249, 3147}, {1495, 9142}, {1503, 44437}, {1513, 45279}, {1545, 5321}, {1546, 5318}, {1576, 44102}, {1634, 8681}, {1885, 6748}, {1942, 1987}, {1989, 14993}, {2079, 37933}, {2174, 2197}, {2252, 8776}, {2269, 22058}, {2351, 44077}, {2386, 21177}, {2790, 44227}, {3133, 12235}, {3148, 19136}, {3163, 18334}, {3164, 41760}, {3199, 13881}, {3580, 18609}, {3815, 30739}, {4383, 21494}, {4558, 35296}, {5181, 11672}, {5191, 9407}, {5305, 16618}, {5412, 44193}, {5413, 44192}, {5486, 43718}, {5702, 46453}, {6509, 26958}, {6663, 13383}, {7113, 7117}, {7202, 43034}, {7297, 14936}, {7493, 7735}, {7669, 18374}, {7736, 15302}, {7745, 34664}, {8623, 18371}, {8962, 32787}, {9027, 9155}, {9145, 32127}, {9220, 39565}, {9300, 43957}, {9475, 38987}, {9723, 40318}, {9822, 35222}, {9969, 23635}, {10313, 37929}, {11188, 37465}, {11328, 29959}, {13567, 46832}, {14060, 37183}, {14569, 15508}, {14575, 41593}, {14581, 35372}, {14984, 44221}, {15073, 37114}, {15355, 39576}, {16237, 44138}, {16238, 45847}, {17907, 41770}, {18122, 34827}, {18362, 36430}, {18877, 21663}, {19118, 44200}, {20775, 32366}, {21639, 39231}, {23357, 44174}, {23980, 35090}, {34104, 39170}, {34394, 36296}, {34395, 36297}, {35370, 44895}, {35486, 39575}, {36982, 41373}, {37778, 41678}, {39176, 44281}, {40350, 44533}, {40423, 46230}, {41758, 45172}
X_3005 — (DE LONGCHAMPS LINE)∩(LEMOINE AXIS)

X3005(DE LONGCHAMPS LINE)∩(LEMOINE AXIS)

Let L denote the line at infinity. Then (Lemoine axis) = (isogonal conjugate of isotomic conjugate of L) and (de Longchamps line) = (isotomic conjugate of isogonal conjugate of L), and X3005 = (Lemoine axis)∩(de Longchamps line). (Randy Hutson, February 10, 2016)
X_3020 — 4th STEVANOVIC POINT

X30204th STEVANOVIC POINT

Centers X3020-X3026 are points on the incircle, as noted by Milorad Stevanovic (Hyacinthos, Dec. 7-8, 2004)
X_3063 — CROSSDIFFERENCE OF X_7 AND X_8

X3063CROSSDIFFERENCE OF X7 AND X8

X3063 is the perspector of the triangle ABC and the tangential triangle of the conic that passes through the points A, B, C, X55, and X56.
X_3067 — HOFSTADTER ELLIPSE INTERSECTION

X3067HOFSTADTER ELLIPSE INTERSECTION

The Hofstadter ellipse E(0) is described at X359. The point other than A, B, and C in which E(0) meets the circumcircle is X3067. As with X359 and X360, this is a transcendental center, with “exposed angles” A, B, C in its coordinates.
X_3078 — DANNEELS POINT OF X_5

X3078DANNEELS POINT OF X5

Let A'B'C' be the cevian triangle of a point X =\(x : y : z\) (trilinears), let L(A) be the line through A parallel to B'C', and define L(B) and L(C) cyclically. The lines L(A), L(B), L(C) form a triangle homothetic to A'B'C', with homothetic center
X_3082 — RADICAL CENTER OF EXTERNAL MALFATTI CIRCLES

X3082RADICAL CENTER OF EXTERNAL MALFATTI CIRCLES

Let (OA) be the I-excircle of triangle IBC, where I = X1, and define (OB) and (OC) cyclically. Let A' be the touchpoint of (OA) and side BC, and define B' and C' cyclically. Let A″ be the insimilicenter of (OB) and (OC), and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X3082. (Randy Hutson, July 11, 2019)
X_3098 — INTERSECTION OF LINES X_3X_6 AND X_69X_74

X3098INTERSECTION OF LINES X3X_6 AND X69X_74

Let AA1A2, BB1B2, CC1C2 be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A' be the midpoint of A1 and A2, and define B' and C' cyclically. Triangle A'B'C' is homothetic to ABC at X3 and perspective to the Trinh triangle at X3098.
X_3102 — INNER VERTEX OF THE BROCARD SQUARE

X3102INNER VERTEX OF THE BROCARD SQUARE

The 1st Brocard point (trilinears \(c/b : a/c : b/a)\) and 2nd Brocard point (\(b/c : c/a : a/b)\) are opposing vertices of a square here called the Brocard square. The other two vertices are X3102 and X3103. Both are on the Brocard axis X3X6, and X3102 is the closer to X3.
X_3108 — DC(X_83)

X3108DC(X83)

For an introduction to the DC and CD mappings, see the discussion just before X2979.