X3638

INNER SODDY-GERGONNE POINT

Write I = X1, U = X7, r = inradius, R = circumradius, and d = 4*\(\sqrt{3}\)*area(ABC). Then distance ratios are given by |IP|/|PU| = 2r(r + 4R)/d and |IQ|/|QU| = -2r(r + 4R)/d
< 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 { 14, 226 }: Central Line L(14, 226) { 57, 1277 }: Central Line L(57, 1277) { 553, 1082 }: Central Line L(553, 1082) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_7 = GERGONNE POINT (0.5903, 0.2769) X_14 = 2nd ISOGONIC CENTER (0.9134, -0.1143) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_1277 = 3rd EVANS PERSPECTOR (0.4702, 0.0672) X_553 = POINT MAIA II (0.5963, 0.2857) X_1082 = POINT ALPHERATZ (0.5483, 0.1908) X_3638 = INNER SODDY-GERGONNE POINT (0.5809, 0.2553) A B C X1 X7 X14 X226 X57 X1277 X553 X1082 X3638 (a, b, c) = (5, 6, 7) | AB = 1

X3638 (INNER SODDY-GERGONNE POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Barycentrics: \((-2 (2 a^{2}-(b+c) a-(b-c)^{2}) S+\sqrt{3} (-a+b+c) \cdot (2 a^{3}-(b+c) a^{2}-(b^{2}-c^{2}) \cdot (b-c))) \cdot (a-b+c) \cdot (a+b-c) : :\)
Search Value (6, 9, 13): \(1.156773119979\)
▸ Incident Central Lines4 lines

{1, 7}, {14, 226}, {57, 1277}, {553, 1082}

▸ Functional Relations3 relations

Notes

In the plane of triangle ABC, let X, Y, Z be the points of contact of the incircle with sidelines BC,CA,AB, and let X',Y',Z' be the harmonic conjugates of X, Y, Z with respect to {B,C}, {C,A}, {A,B}, respectively. It is well known that AX, BY, CZ concur in the Gergonne point, X7, and that X',Y',Z' lie on the Gergonne line. The circles with centers X',Y',Z' passing respectively through X, Y, Z are orthogonal to the incircle and to the Gergonne line, forming a coaxal system with the Gergonne line as line of centers. The radical axis is the Soddy line, which passes through X1 and X7 and is the line of centers of the orthogonal coaxal system, which includes the Soddy circles. Among the points on this line are the two limiting points (point-circles of the system): X3638 and X3639. The former lies between X7 and the inner Soddy point, X175. (Richard Guy, September 2, 2011)

Coordinates and properties for the points P=X3638 and Q=X3639 were found by Peter Moses (September 3, 2011):