X1115–X1165

Morley-Taylor-Marr, Hofstadter, Ehrmann & Star Centers

Morley-Taylor-Marr centers, Hofstadter points, the Ehrmann congruent squares point (X1144), Grebe triangle centers, and star-named points (Aldebaran, Algjebbah, Altair, Altais).
< 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_1115 = EXTERIOR-ANGLE CURVATURE CENTROID (0.5140, 0.1692) X_1116 = CENTER OF THE LESTER CIRCLE (0.7561, 0.0883) X_1117 = POINT ALRESCHA (0.9978, 0.0022) X_1118 = 1st HATZIPOLAKIS PERSPECTOR (0.6247, 0.5521) X_1120 = -BLAIKIE TRANSFORM OF X(1) (1.2041, 0.2000) X_1121 = -BLAIKIE TRANSFORM OF X(2) (0.8358, -0.1143) X_1131 = ARCTAN(2) KIEPERT POINT (0.5904, 0.2888) X_1132 = ARCTAN(-2) KIEPERT POINT (0.8864, -0.3774) X_1136 = 5th MORLEY-TAYLOR-MARR CENTER (0.6158, 0.4039) X_1138 = ISOGONAL CONJUGATE OF X(399) (-0.2714, 0.2752) X_1140 = INNER PENTAGON POINT (0.5248, 1.3312) X_1141 = GIBERT POINT (-0.0724, -0.2027) X_1145 = 3rd EHRMANN POINT (0.3061, 0.2999) X_1147 = ISOGONAL CONJUGATE OF X(847) (0.8329, 0.3027) X_1160 = CIRCUMCENTER OF OUTER GREBE TRIANGLE (0.1258, -0.5160) X_1161 = CIRCUMCENTER OF INNER GREBE TRIANGLE (0.8742, 0.7201) A B C X3 X1 X1115 X1116 X1117 X1118 X1120 X1121 X1131 X1132 X1136 X1138 X1140 X1141 X1145 X1147 X1160 X1161 (a, b, c) = (5, 6, 7) | AB = 1

X1115–X1165 (Morley-Taylor-Marr, Hofstadter, Ehrmann & Star Centers) — 16 representative centers (from 51 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Centers 1115-1150 (Centers X1115-X1150, Jan 10, 2003)

were added to ETC on 1/10/03.

X_1115 — EXTERIOR-ANGLE CURVATURE CENTROID

X1115EXTERIOR-ANGLE CURVATURE CENTROID

X1115 is the center of mass of a point-mass system obtained by placing at vertex :g:`A <A*>` a mass equal to the magnitude of the exterior angle (that’s π - :g:`A <A*>`) at :g:`A <A*>`, and cyclically for :g:`B <B*>` and :g:`C <C*>`. ( Peter Scales , Hyacinthos #5528, 5/22/02) This description is also found in Honsberger , Episodes in Nineteenth and Twentieth Century Eulidean Geometry , p. 120, where it is mistakenly attributed to the Steiner point, X99.
X_1116 — CENTER OF THE LESTER CIRCLE

X1116CENTER OF THE LESTER CIRCLE

The appearance of i in the following list means that Xi lies on the Lester circle: 3, 5, 13, 14, 1117, 5671, 14854, 15475, 15535, 15536, 15537, 15538, 15539, 15540, 15541, 15542, 15543, 15544, 15545, 15546, 15547, 15548, 15549, 15550, 15551, 15552, 15553, 15554, 15555, 34365
X_1117 — POINT ALRESCHA

X1117POINT ALRESCHA

X1117 on the Lester circle, the cubic K060, and these lines: {5,3470}, {30,5671}, {265,13582}, {11071,11581}
X_1118 — 1st HATZIPOLAKIS PERSPECTOR

X11181st HATZIPOLAKIS PERSPECTOR

Let TaTbTc be the intouch triangle of ABC. Let CA be the point other than Tc in which the perpendicular to BC from Tc meets the incircle, let BA be the point other than Tb in which the perpendicular to BC from Tb meets the incircle, and let A0 be the point of intersection of lines BCA and CBA. Define B0 and C0 cyclically. Then triangle A0B0C0 is perspective to ABC, and the perspector is X1118. (Antreas Hatzipolakis, #5321, 4/30/02)
X_1119 — 2nd HATZIPOLAKIS PERSPECTOR

X11192nd HATZIPOLAKIS PERSPECTOR

Let triangle A0B0C0 be as defined for X1118. Let A1 be the orthogonal projection of A0 onto line BC, and define B1 and C1 cyclically. Then triangle A1B1C1 is perspective to ABC, and the perspector is X1119. (Antreas Hatzipolakis, #5321, 4/30/02)
X_1122 — 1st GRINBERG POINT

X11221st GRINBERG POINT

Let AB be the touchpoint of the A-excircle and line AB, let AC be the touchpoint of the A-excircle and line AC, and let MA be the midpoint of segment ABAC. Define MB and MC cyclically. Let A', B', C' be the touchpoints of the incircle with lines BC, CA, AB, respectively. The triangles MAMBMC and A'B'C' are perspective, and the perspector is X1122. (Darij Grinberg, 12/28/02)
X_1123 — PAASCHE POINT

X1123PAASCHE POINT

Let D and E be the congruent circles each tangent to the other and to line BC, with D also tangent to line AB and E also tangent to line CA, meeting in a point A' lying outside triangle ABC. Define B' and C' cyclically. Then A'B'C' is perspective to ABC, and the perspector is X1123. See
X_1131 — ARCTAN(2) KIEPERT POINT

X1131ARCTAN(2) KIEPERT POINT

To construct the Vecten point, X485, squares are erected outward on the sides of ABC. If A', B', C' are the centers of these squares, then triangle A'B'C' is perspective to ABC with perspector X485. Now let A″ be the midpoint of the side of the A-square that does not touch line BC, and define B″ and C″ cyclically. Then triangle A″B″C″ is perspective to ABC with perspector X1131. Angle(A″BC) = angle(B″CA) = angle(C″AB), so that X1131 lies on the Kiepert hyperbola. Here, the common angle is arctan(2). (Darij Grinberg 9/22/02)
X_1132 — ARCTAN(-2) KIEPERT POINT

X1132ARCTAN(-2) KIEPERT POINT

X1132 is constructed in the manner for X1131, using squares erected inward, so that the three equal angles have common measure arctan(-2), and X1132 lies on the Kiepert hyperbola. (Darij Grinberg 9/22/02)
X_1133 — BURGESS POINT

X1133BURGESS POINT

Rotate line BC about B away from A through angle B/3, and rotate line BC about C away from A through angle C/3; let A' be the point in which the two rotated lines meet. Define B' and C' cyclically. Let A″ be the point of intersection of lines BC' and B'C, and define B″ and C″ cyclically. The lines AA', BB', CC' concur in X357, and AA″, BB″, CC″ concur in X358. The first of these with reference to triangle A'B'C' is X1133; i.e., X1133 = X357-of-A'B'C'.
X_1139 — OUTER PENTAGON POINT

X1139OUTER PENTAGON POINT

Let A' be the outermost vertex of the regular pentagon erected outward on side BC of ABC. Define B' and C' cyclically. Then triangle A'B'C' is perspective to ABC, and the perspector is X1139. (Steve Sigur and Antreas Hatzipolakis, #5246, 12/31/02; generalizations to n-gons, for odd n, by Milorad R. Stevanovic, #5253, 5256)
X_1140 — INNER PENTAGON POINT

X1140INNER PENTAGON POINT

Let A' be the innermost vertex of the regular pentagon erected inward on side BC of ABC. Define B' and C' cyclically. Then triangle A'B'C' is perspective to ABC, and the perspector is X1140. See references at X1140.
X_1141 — GIBERT POINT

X1141GIBERT POINT

In Jean-Pierre Ehrmann and Bernard Gibert, “Special Isocubics,” downloadable from Cubics in the Triangle Plane, the point X1141 is labeled E, barycentrics are given, and it is established that this point also lies on the line X5-to-X110 [listed below as 5,49], two other cubics, and the hyperbola that passes through the points A, B, C, X4, X5.
X_1143 — 2nd MALFATTI-RABINOWITZ POINT

X11432nd MALFATTI-RABINOWITZ POINT

Along each side of ABC there is a segment that is a common tangent to two of the three Malfatti circles of ABC. Let A', B', C' be the midpoints of these respective segments. Then triangle A'B'C' isperspective to ABC, and the perspector is X1143. (Stanley Rabinowitz, #4611, 12/29/01) For coordinates, see Paul Yiu, #4615, 12/30/01, and
X_1144 — EHRMANN CONGRUENT SQUARES POINT

X1144EHRMANN CONGRUENT SQUARES POINT

Suppose that P is a point inside triangle ABC. Let \(S_{A}\) be the square inscribed in triangle PBC, having two vertices on segment BC, one on PB, and one on PC. Define \(S_{B}\) and \(S_{C}\) cyclically. Then X1144 is the unique choice of P for which the three squares are congruent. The function L(a,b,c) is symmetric, homogeneous of degree 1, and satisfies 0 < L(a,b,c) < min{a,b,c}. Also, X1144 lies on the hyperbola {A, B, C,X1,X6}; indeed, X1144 lies on the open arc from X1 to the vertex of ABC opposite the shortest side. L(a,b,c) is the common length of the sides of the three squares. (Jean-Pierre Ehrmann, 12/16/01)
X_1145 — 3rd EHRMANN POINT

X11453rd EHRMANN POINT

Let Ja,Jb,Jc be the respective excenters of ABC, and let AB be the projection of A on JaJb, let AC be the projection of A on JaJc, and define BC, BA, CA, CB cyclically. The Euler lines of the three triangles JaABAC, JbBCBA, JcCACB concur in X1145. Also, X1145 is X974 of the excentral triangle. (Analogously, X442 is X973 of the excentral triangle; see the note at X442.) Jean-Pierre Ehrmann (#4200, 10/24/01)
X_1148 — YIU-HATZIPOLAKIS POINT

X1148YIU-HATZIPOLAKIS POINT

Suppose LA, LB, LC are lines through a point P, respectively perpendicular to sidelines BC, CA, AB. Let AB be the point where LA meets AB, and let AC be the point where LA meets AC. Define BC, BA, CA, CB cyclically. Then X1148 is the point P, which satisfies
X_1151 — ISOGONAL CONJUGATE OF X_1131

X1151ISOGONAL CONJUGATE OF X1131

X1151 is the radical center of the Lucas circles, the incenter of the Lucas central triangle, and the perspector of triangle ABC and the Lucas inner triangle.
X_1153 — CENTER OF THE VAN LAMOEN CIRCLE

X1153CENTER OF THE VAN LAMOEN CIRCLE

A triangle is divided by its three medians into 6 smaller triangles. The circumcenters of these smaller triangles are concyclic. Their circle, the Van Lamoen circle, is introduced in
X_1155 — SCHRÖDER POINT

X1155SCHRÖDER POINT

Let XYZ be the intouch triangle of ABC; i.e., the pedal triangle of the incenter, I. The circles AIX, BIY, CIZ concur in two points. One of them is I; the other is X1155. This result is obtain by inversion in
X_1157 — INVERSE-IN-CIRCUMCIRCLE OF X_54

X1157INVERSE-IN-CIRCUMCIRCLE OF X54

For any point X, let XA be the reflection of X in sideline BC, and define XB and XC cyclically. Then X1157 is the unique point X for which the lines AXA, BXB, CXC concur on the circumcircle; the point of concurrence is X1141.