X351–X400

Parry Circle, Morley/Yff Centers & Pedal Centroids

The Parry circle center, Morley trisector centers, Yff equal-parallelians, trisected-perimeter, and equilateral-cevian points, Kenmotu point, and centroids of pedal and cevian triangles.
< 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_351 = CENTER OF THE PARRY CIRCLE (1.3537, 1.1619) X_354 = WEILL POINT (0.5823, 0.2533) X_359 = HOFSTADTER ONE POINT (0.5571, 0.2061) X_361 = -CEVA CONJUGATE OF X(1) (0.6578, 0.3310) X_369 = 1st TRISECTED PERIMETER POINT (0.6760, -0.0539) X_372 = {X(3),X(6)}-HARMONIC CONJUGATE OF X(371) (0.7148, 0.4568) X_376 = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2) (0.4626, 0.0042) X_378 = REFLECTION OF X(22) IN X(3) (0.5857, 0.3262) X_382 = REFLECTION OF CIRCUMCENTER IN ORTHOCENTER (0.7245, 0.6894) X_391 = INTERSECTION OF LINES X(2,6) AND X(8,9) (0.4857, 0.1443) X_393 = -CROSS CONJUGATE OF X(4) (0.6247, 0.5263) X_394 = -CEVA CONJUGATE OF X(3) (0.3778, 0.0284) X_395 = MIDPOINT OF X(14) AND X(16) (1.3794, 1.1048) X_399 = PARRY REFLECTION POINT (0.0032, 0.9936) X_400 = YFF-MALFATTI POINT (0.3372, 0.0569) A B C X3 X1 X351 X354 X359 X361 X369 X372 X376 X378 X382 X391 X393 X394 X395 X399 X400 (a, b, c) = (5, 6, 7) | AB = 1

X351–X400 (Parry Circle, Morley/Yff Centers & Pedal Centroids) — 15 representative centers (from 50 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X_351 — CENTER OF THE PARRY CIRCLE

X351CENTER OF THE PARRY CIRCLE

X351 is the center of the Parry circle introduced in TCCT (Art. 8.13) as the circle that passes through Xi for I = 2, 15, 16, 23, 110, 111, 352, 353.
X_353 — INVERSE-IN-BROCARD-CIRCLE OF X_111

X353INVERSE-IN-BROCARD-CIRCLE OF X111

Let AaBaCa, AbBbCb, AcBcCc be the A-, B- and C-anti-altimedial triangles. Let A' be the trilinear product Aa*Ab*Ac, and define B' and C' cyclically. Triangle A'B'C' is the anticomplementary triangle of the 1st Brocard triangle, and is similar to the anti-orthocentroidal triangle, with similitude center X353. (Randy Hutson, November 2, 2017)
X_355 — FUHRMANN CENTER

X355FUHRMANN CENTER

X355 = the center of the Fuhrmann circle, defined as the circumcircle of the Fuhrmann triangle A″B″C″, where A″ is obtained as follows: let A' be the midpoint of the circumcircle-arc having endpoints B and C and not containing A; then A″ is the reflection of A' in line BC. Vertices B″ and C″ are obtained cyclically. (Other constructions of A″, and hence the Fuhrmann triangle, follow: (1) Let IA be the reflection of X1 in BC; then A″ is the circumcenter of IABC. (2) Let JA be the reflection of the A-excenter in BC; then A″ is the circumcenter of JABC.
X_356 — MORLEY CENTER

X356MORLEY CENTER

X356 is the centroid of the Morley equilateral triangle. For a discussion of the theorem and extensive list of references, see
X_359 — HOFSTADTER ONE POINT

X359HOFSTADTER ONE POINT

The Hofstadter ellipse E(1/2), given by \(x^{2}\) + \(y^{2}\) + \(z^{2}\) - 2yz - 2xz - 2xy = 0, has center X37 and passes through Xi for these i: 244, 678, 2310, 2632, 2638, 2643, 3248, 4094, 4117, 10501, 23063, 24012.
X_360 — HOFSTADTER ZERO POINT

X360HOFSTADTER ZERO POINT

The limit of this point as r approaches 0 is X360. The two Hofstadter points, X359 and X360 are examples of transcendental triangle centers, since they have no trilinear or barycentric representation using only algebraic functions of a,b,c (or sin A, sin B, sin C).
X_365 — SQUARE ROOT POINT

X365SQUARE ROOT POINT

For a construction of X365, see the note at X2, which provides for a construction barycentric square roots which one can easily extend to a construction for trilinear square roots.
X_368 — EQUI-BROCARD CENTER

X368EQUI-BROCARD CENTER

The center X for which the triangle XBC, XCA, XAB have equal Brocard angles. Peter Yff proved that X368 lies on the self-isogonal conjugate cubic with trilinear equation f(a,b,c)u + f(b,c,a)v + f(c,a,b)w = 0, where \(f(a,b,c) = bc(b^{2} - c^{2})\) and, for variable \(x : y : z\), the cubics u, v, w are given by u(x,y,z) = \(x(y^{2}\) + \(z^{2})\), v = u(y,z,x), w = u(z,x,y).
X_369 — 1st TRISECTED PERIMETER POINT

X3691st TRISECTED PERIMETER POINT

There exist points A', B', C' on segments BC, CA, AB, respectively, such that AB' + AC' = BC' + BA' = CA' + CB' = (a + b + c)/3, and the lines AA', BB', CC' concur in X369. Near the end of the 20th century, Yff found trilinears for X369 in terms of the unique real root, r, of the cubic polynomial
X_371 — KENMOTU POINT (CONGRUENT SQUARES POINT)

X371KENMOTU POINT (CONGRUENT SQUARES POINT)

There exist three congruent squares U, V, W positioned in ABC as follows: U has opposing vertices on segments AB and AC; V has opposing vertices on segments BC and BA; W has opposing vertices on segments CA and CB, and there is a single point common to U, V, W. The common point, X371, may have first been published in Kenmotu’s Collection of Sangaku Problems in 1840, indicating that its first appearance may have been anonymously inscribed on a wooden board hung up in a Japanese shrine or temple. (The Kenmotu configuration uses only half-squares; i.e., isosceles right triangles). Trilinears were found by John Rigby.
X_381 — MIDPOINT OF X_2 AND X_4

X381MIDPOINT OF X2 AND X4

X381 is the point QA-P14 (Centroid of the Morley Triangle) of quadrangle ABCX2 (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/42-qa-p14.html)
X_395 — MIDPOINT OF X_14 AND X_16

X395MIDPOINT OF X14 AND X16

X395 is the center of the (equilateral) pedal triangle of X16, as well as the circumcenter of the pedal triangle of X14.
X_396 — MIDPOINT OF X_13 AND X_15

X396MIDPOINT OF X13 AND X15

X396 is the center of the (equilateral) pedal triangle of X15, as well as the circumcenter of the pedal triangle of X13.
X_399 — PARRY REFLECTION POINT

X399PARRY REFLECTION POINT

Let L, M, N be lines through A, B, C, respectively, parallel to the Euler line. Let L' be the reflection of L in sideline BC, let M' be the reflection of M in sideline CA, and let N' be the reflection of N in sideline AB. The lines L', M', N' concur in X399, as proved in
X_400 — YFF-MALFATTI POINT

X400YFF-MALFATTI POINT

In 1997, Yff considered the configuration for the 1st Ajima-Malfatti point, X179. He proved that the same tangencies are possible in another way if the circles are not required to lie inside ABC. With tangency points labeled as before, the lines AA', BB', CC' concur in X400.