X140–X190

Derived Triangle Centers, Isoscelizers & Ajima-Malfatti Points

Complements, anticomplements, and classical centers evaluated on the medial, orthic, tangential, and excentral triangles, together with congruent isoscelizer, equal detour, and Ajima-Malfatti points.
< 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_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_143 = NINE-POINT CENTER OF ORTHIC TRIANGLE (0.6358, 0.3534) X_146 = REFLECTION OF X(20) IN X(110) (0.1155, 1.2873) X_147 = TARRY POINT OF ANTICOMPLEMENTARY TRIANGLE (-0.1660, 1.0561) X_148 = STEINER POINT OF ANTICOMPLEMENTARY TRIANGLE (1.3904, -0.2646) X_149 = REFLECTION OF X(20) IN X(104) (1.3265, -0.3333) X_150 = REFLECTION OF X(20) IN X(103) (1.1837, -0.4499) X_152 = REFLECTION OF X(20) IN X(101) (0.0408, 1.2414) X_153 = REFLECTION OF X(20) IN X(100) (-0.1020, 1.1248) X_155 = EIGENCENTER OF ORTHIC TRIANGLE (1.1658, 0.5033) X_156 = -OF-TANGENTIAL-TRIANGLE (0.6520, -0.1707) X_157 = -OF-TANGENTIAL-TRIANGLE (0.5266, 0.3812) X_160 = -OF-TANGENTIAL-TRIANGLE (0.4841, -0.0647) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4762, 0.0583) X_173 = CONGRUENT ISOSCELIZERS POINT (0.5924, 0.2741) X_190 = YFF PARABOLIC POINT (0.0748, 0.3999) A B C X3 X1 X140 X143 X146 X147 X148 X149 X150 X152 X153 X155 X156 X157 X160 X165 X173 X190 (a, b, c) = (5, 6, 7) | AB = 1

X140–X190 (Derived Triangle Centers, Isoscelizers & Ajima-Malfatti Points) — 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 X140-X170 (Centers of central triangles)

Centers in this section, by index:

113- 127, 140- 143: centers of the medial triangle
128- 139: centers of the orthic triangle
144- 153: centers of the anticomplementary triangle
154- 157, 159- 163: centers of the tangential triangle
164- 170: centers of the excentral triangle
X_142 — COMPLEMENT OF X_9

X142COMPLEMENT OF X9

X142 is the {X2,X7}-harmonic conjugate of X9. For a list of other harmonic conjugates, click Tables at the top of this page.
X_143 — NINE-POINT CENTER OF ORTHIC TRIANGLE

X143NINE-POINT CENTER OF ORTHIC TRIANGLE

X143 is the QA-P13 center (Nine-point Center of the QA-Diagonal Triangle) of quadrangle ABCX4 (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/41-qa-p13.html). Also, X143 is the QA-P22 center (Midpoint QA-P1 and QA-P20) of quadrangle ABCX4 (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/50-qa-p22.html)
X_145 — ANTICOMPLEMENT OF NAGEL POINT

X145ANTICOMPLEMENT OF NAGEL POINT

Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the same side of BC as A. Define OB, OC cyclically. Then X145 is the trilinear pole of the line of the exsimilicenters (the Monge line) of OA, OB, OC. See the reference at X1001.
X_146 — REFLECTION OF X_20 IN X_110

X146REFLECTION OF X20 IN X110

X146 is the {X74,X113}-harmonic conjugate of X2. For a list of harmonic conjugates, click Tables at the top of this page.
X_149 — REFLECTION OF X_20 IN X_104

X149REFLECTION OF X20 IN X104

X149 is the {X11,X100}-harmonic conjugate of X2. For a list of harmonic conjugates, click Tables at the top of this page.
X_154 — X_3-CEVA CONJUGATE OF X_6

X154-CEVA CONJUGATE OF X6

X154 is the {X26,X156}-harmonic conjugate of X155. For a list of harmonic conjugates of X154, click Tables at the top of this page.
X_156 — X_5-OF-TANGENTIAL-TRIANGLE

X156-OF-TANGENTIAL-TRIANGLE

X156 is the {X154,X155}-harmonic conjugate of X26. For a list of harmonic conjugates, click Tables at the top of this page.
X_173 — CONGRUENT ISOSCELIZERS POINT

X173CONGRUENT ISOSCELIZERS POINT

Let PB on sideline AC and QC be equidistant from A, so that APBQC is an isosceles triangle. The line PBQC is called an isoscelizer. The lines PBQC, PCQA, PAQB concur in X173. (P. Yff, unpublished notes, 1989)
X_174 — YFF CENTER OF CONGRUENCE

X174YFF CENTER OF CONGRUENCE

In notes dated 1987, Yff raises this question concerning certain triangles lying within ABC: can isoscelizers (defined at X173), PBQC, PCQA, PAQB, be constructed so that, on putting
X_175 — ISOPERIMETRIC POINT

X175ISOPERIMETRIC POINT

A point X is defined as an isoperimetric point of triangle ABC if |XB| + |XC| + |BC| = |XC| + |XA| + |CA| = |XA| + |XB| + |AB|. Veldkamp established that X = X175, uniquely, for some triangles ABC, but the conditions he gives are not correct. Hajja and Yff proved that the condition tan(A/2) + tan(B/2) + tan(C/2) < 2 is necessary and sufficient. See also X176 and the 1st and 2nd Eppstein points, X481, X482.
X_176 — EQUAL DETOUR POINT

X176EQUAL DETOUR POINT

A point X is defined as a point of equal detour of triangle ABC if |XB| + |XC| - |BC| = |XC| + |XA| - |CA| = |XA| + |XB| - |AB|. Veldkamp established that X = X176 for some triangles ABC, but the conditions he gives are not correct. Hajja and Yff proved that the condition tan(A/2) + tan(B/2) + tan(C/2) < 2 is necessary and sufficient for the existence of exactly two points of equal detour and that the condition tan(A/2) + tan(B/2) + tan(C/2) = 2 is necessary and sufficient for the existence of exactly one point of equal detour. Yff found that X176 is also is the center of the inner Soddy circle. See also X175 and the 1st and 2nd Eppstein points, X481, X482.
X_179 — 1st AJIMA-MALFATTI POINT

X1791st AJIMA-MALFATTI POINT

The famous Malfatti Problem is to construct three circles O(A), O(B), O(C) inside ABC such that each is externally tangent to the other two, O(A) is tangent to lines AB and AC, O(B) is tangent to BC and BA, and O(C) is tangent to CA and CB. Let A' = O(B)∩O(C), B' = O(C)∩O(A), C' = O(A)∩O(B). The lines AA',BB',CC' concur in X179. Trilinears are found in Yff’s unpublished notes. See also the Yff-Malfatti Point, X400, having trilinears \(\csc^{4}(A/4)\) : \(\csc^{4}(B/4)\) : \(\csc^{4}(C/4)\), and the references for historical notes.
X_180 — 2nd AJIMA-MALFATTI POINT

X1802nd AJIMA-MALFATTI POINT

Let A″,B″,C″ be the excenters of ABC, and let A',B',C' be as in the construction of X179. The lines A'A″,B'B″,B'B″ concur in X180. Trilinears are found in Yff’s unpublished notes. See X179.
X_181 — APOLLONIUS POINT

X181APOLLONIUS POINT

Let O(A),O(B),O(C) be the excircles. Apollonius’s Problem includes the construction of the circle O tangent to the three excircles and encompassing them. (The circle is called the Apollonius circle.) Let A' = O∩O(A), B'=O∩O(B), C'=O∩O(C). The lines AA',BB',CC' concur in X181. Yff derived trilinears in 1992.
X_182 — MIDPOINT OF BROCARD DIAMETER

X182MIDPOINT OF BROCARD DIAMETER

X182 is the midpoint of the Brocard diameter (the segment X3-to-X6); also the center of the 1st Lemoine circle, and the center of the Brocard circle.
X_184 — INVERSE OF X_125 IN THE BROCARD CIRCLE

X184INVERSE OF X125 IN THE BROCARD CIRCLE

X184 is the homothetic center of triangles ABC and A'B'C', the latter defined as follows: let B1 and C1 be the points where the perpendicular bisector BC meets sidelines CA and AB, and cyclically define C2, A2; A3, B3. Then A'B'C' is formed by the perpendicular bisectors of segments B1C1, C2A2, A3B3. (Fred Lang, Hyacinthos #1190)
X_190 — YFF PARABOLIC POINT

X190YFF PARABOLIC POINT

In unpublished notes, Yff has studied the parabola tangent to sidelines BC, CA, AB and having focus X101. If A',B',C' are the respective points of tangency, then the lines AA', BB', CC' concur in X190.