X481

1st EPPSTEIN POINT

Let S be the inner Soddy circle and \(S_{A}\), \(S_{B}\), \(S_{C}\) the Soddy circles tangent to S. Let Ia = S∩\(S_{A}\), Ea = \(S_{B}\)∩\(S_{C}\), and determine Ib, Ic, Eb, Ec cyclically. Then X481 is the point of concurrence of lines IA-to-EA, IB-to-EB, IC-to-EC.
< 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) { 1, 7 }: Soddy Line { 226, 485 }: Central Line L(226, 485) { 174, 1127 }: Central Line L(174, 1127) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_79 = ISOGONAL CONJUGATE OF X(35) (0.5976, 0.3019) X_482 = 2nd EPPSTEIN POINT (0.5835, 0.2611) X_1 = INCENTER (0.5714, 0.2333) X_7 = GERGONNE POINT (0.5903, 0.2769) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_485 = VECTEN POINT (0.5773, 0.2578) X_174 = YFF CENTER OF CONGRUENCE (0.5813, 0.2549) X_1127 = 1st DE VILLIERS POINT (0.5779, 0.2467) X_481 = 1st EPPSTEIN POINT (0.6148, 0.3335) A B C X79 X482 X1 X7 X226 X485 X174 X1127 X481 (a, b, c) = (5, 6, 7) | AB = 1

X481 (1st EPPSTEIN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1 - 2 \sec A/2 \cos B/2 \cos C/2 : :\)
Search Value (6, 9, 13): \(-0.019071250304\)
▸ Alternate Coordinates1 form

  • Trilinears: \(1 - 4(\operatorname{area})/[a(b + c - a)] : :\) (E. Brisse, 3/20/01)

▸ Linear Combinations3 combinations

  • X481 = \(s\) X1 - \((r + 4R)\) X7

  • X481 = A / \(R_{a}\) + B / \(R_{b}\) + C / \(R_{c}\) - X1 / \(r\), where \(R_{a}\), \(R_{b}\), \(R_{c}\) = radii of Soddy circles

  • X481 = A / \(R_{a}\) + B / \(R_{b}\) + C / \(R_{c}\) + X175 / \(R_{s}'\), where \(R_{a}\), \(R_{b}\), \(R_{c}\) = radii of Soddy circles, and \(R_{s}'\) = radius of outer Soddy circle

▸ Incident Central Lines3 lines

{1, 7}, {174, 1127}, {226, 485}

▸ Functional Relations6 relations

Notes

David Eppstein, “Tangent spheres and triangle centers,”American Mathematical Monthly, 108 (2001) 63-66.

Let S' be the outer Soddy center, X175. X481 is the point of concurrence of the Soddy lines of BS'C, CS'A, and AS'B. (Randy Hutson, September 14, 2016)

Let La be the line tangent to the outer Soddy circle at the touchpoint with the A-Soddy circle. Define Lb and Lc cyclically. Let A' = Lb∩Lc and define B' and C’cyclically. Triangle A'B'C', here introduced as the outer Soddy tangential triangle, is perspective to ABC, and the perspector is X481. (Randy Hutson, September 14, 2016)