X396

MIDPOINT 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.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 13, 15 }: Central Line L(13, 15) { 2, 6 }: Centroid–Symmedian Line { 5, 14 }: Central Line L(5, 14) { 62, 140 }: Central Line L(62, 140) { 3, 397 }: Central Line L(3, 397) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_13 = 1st ISOGONIC CENTER (FERMAT POINT, TORRICELLI POINT) (0.5879, 0.2819) X_15 = 1st ISODYNAMIC POINT (0.5519, 0.1878) X_2 = CENTROID (0.5374, 0.2000) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_62 = ISOGONAL CONJUGATE OF X(18) (0.6446, 0.3409) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_397 = CROSSPOINT OF X(4) AND X(17) (0.6029, 0.2975) X_396 = MIDPOINT OF X(13) AND X(15) (0.5699, 0.2349) A B C X13 X15 X2 X6 X5 X14 X62 X140 X3 X397 X396 (a, b, c) = (5, 6, 7) | AB = 1

X396 (MIDPOINT OF X13 AND X15) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\cos(B - C) + 2 \cos(A - \pi /3) : :\)
Search Value (6, 9, 13): \(1.589194586230\)
▸ Incident Central Lines15 lines

{2, 6}, {3, 397}, {5, 14}, {13, 15}, {16, 549}, {39, 619}, {53, 473}, {62, 140}, {115, 531}, {187, 530}, {203, 495}, {216, 466}, {465, 577}, {532, 618}, {533, 623}

▸ Functional Relations13 relations

Notes

X396 is the centroid of the points Ab, Ac, Bc, Ba, Ca, Cb in the construction of the 3rd Fermat-Dao equilateral triangle; see X16267. Also, X396 is the center of the ellipse passing through these six points. (Randy Hutson, March 14, 2018)

Let F = X13, the 1st Fermat point. Let Ab be the point on CA and Ac the point on AB such that FAbAc is an equilateral triangle. Let A' be the midpoint of FAbAc, and define B' and C' cyclically. The triangle A'B'C', here named the 7th Fermat-Dao equilateral triangle, is perspective to ABC. The circumcircle of A'B'C' is the pedal circle of X13 and (X15), and the centroid of A'B'C' is X396. The perspector of ABC and A'B'C' is X2, and the centroid of A'B'C' is X396.

X396 is the pole of the Euler line wrt every conic passing through X13, X15, X17 and X61. (Randy Hutson, January 17, 2020)