X1478–X1511

Johnson-Yff Circle, Perspectors & Star-Named Centers

The Johnson-Yff circle center, tangential-of-tangential perspectors, star-named centers (Bellatrix, Betelgeuse, Biham), and Napoleon and Fermat crosssums.
< 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_1478 = CENTER OF JOHNSON-YFF CIRCLE (0.5842, 0.2843) X_1481 = POINT ASTEROPE (0.6161, 0.3645) X_1484 = REFLECTION OF X(5) IN X(11) (0.9133, -0.1156) X_1485 = PERSPECTOR OF ABC AND TANGENTIAL-OF-TANGENTIAL TRIANGLE (0.7356, 0.1420) X_1487 = NAPOLEON CEVAPOINT (0.4925, 0.1879) X_1490 = ∩X(3)X(9) (0.3929, -0.4775) X_1492 = COLUMBUS POINT (0.2205, 0.5198) X_1494 = ISOTOMIC CONJUGATE OF X(30) (0.6185, -0.1856) X_1495 = CROSSSUM OF X(2) AND X(30) (0.8194, 1.4855) X_1499 = POINT BIHAM (0.1078, 0.5997) X_1502 = ISOGONAL CONJUGATE OF X(1501) (0.3682, 0.0896) X_1503 = POINT ARKAB (0.4818, -0.3843) X_1505 = EXSIMILICENTER OF MOSES AND 2nd LEMOINE CIRCLES (0.4682, 0.0496) X_1511 = FERMAT CROSSSUM (0.3758, 0.3250) A B C X3 X1 X1478 X1481 X1484 X1485 X1487 X1490 X1492 X1494 X1495 X1499 X1502 X1503 X1505 X1511 (a, b, c) = (5, 6, 7) | AB = 1

X1478–X1511 (Johnson-Yff Circle, Perspectors & Star-Named Centers) — 14 representative centers (from 34 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X_1478 — CENTER OF JOHNSON-YFF CIRCLE

X1478CENTER OF JOHNSON-YFF CIRCLE

The Johnson Circle Theorem is the fact that if three congruent circles intersect in a point, then the circle passing through the other three intersections is congruent to them. This fourth circle is the Johnson circle of the three given circles. There are three congruent circles each tangent to two sides of triangle ABC. Peter Yff proved that their Johnson circle has center X1478. The circle is here named the Johnson-Yff circle of the triangle.
X_1487 — NAPOLEON CEVAPOINT

X1487NAPOLEON CEVAPOINT

Let N denote the nine-point center, X5. Let NA = N-of-triangle NBC, and define NB and NC cyclically. Triangle NANBNC is perspective to ABC, and X1487 is the perspector. X1487 is the cevapoint of the Napoleon points, X17 and X18. (Coordinates found by Paul Yiu.)
X_1498 — REFLECTION OF X_64 IN X_3

X1498REFLECTION OF X64 IN X3

Let (Ba, Ca) be the points where BC is cut by the parallels through A to the b- and c- altitudes, respectively, and define (Cb, Ab), (Ac, Bc) cyclically. Then the perpendicular bisectors of (Ba, Ca), (Cb, Ac) and (Ac, Bc) concur at X1498. (Hyacinthos #62, #72, #73)
X_1500 — INSIMILICENTER OF MOSES CIRCLE AND INCIRCLE

X1500INSIMILICENTER OF MOSES CIRCLE AND INCIRCLE

The circle having center X39 and radius R tan ω sin 2ω, where R denotes the circumradius of triangle ABC, is here introduced as the Moses circle. It is tangent to the nine-point circle at X115, and its internal and external centers of similitude with the incircle are X1500 and X1015, respectively. (Peter J. C. Moses, 5/29/03)
X_1503 — POINT ARKAB

X1503POINT ARKAB

As the isogonal conjugate of a point on the circumcircle, X1503 lies on the line at infinity.
X_1507 — 1st MORLEY-GIBERT POINT

X15071st MORLEY-GIBERT POINT

X1507 and X1508 are centers on Morley cubic, which also passes through X356, X357, X358; see Bernard Gibert’s site.
X_1508 — 2nd MORLEY-GIBERT POINT

X15082nd MORLEY-GIBERT POINT

X1508 is the perspector of the excentral tiangle and the 1st Morley adjunct triangle. (César Lozada, January 18, 2015)