X(7) = GERGONNE POINT

Trilinears

\(bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)\)

\(sec2(A/2) : sec2(B/2) : sec2(C/2)\)

\(1/(tan(B/2) + tan(C/2)) : 1/(tan(C/2) + tan(A/2)) : 1/(tan(A/2) + tan(B/2))\)

\((bc - SA)/a : (ca - SB)/b : (ab - SC)/c\)

\((1 - cos A)a-2 : :\)

Barycentrics

\(1/(b + c - a) : 1/(c + a - b) : 1/(a + b - c)\)

\(tan A/2 : tan B/2 : tan C/2\)

\(bc - SA : ca - SB : ab - SC\)

\(ra : rb : rc, where ra, rb, rc are the exradii\)

Notes

Let A’, B’, C’ denote the points in which the incircle meets the sides BC, CA, AB, respectively. The lines AA’, BB’, CC’ concur in X(7).

An exarc circle is a circle tangent to two sides of a triangle ABC and externally tangent to the circumcircle of ABC. The point whose distance to the sides BC, CA, AB are proportional to the respective radii of the exarc circles is X(279). The point having distances to the sides proportional to the radii of the inarc circles is X(7). See Martin Lukarevski, “Exarc radii and the Finsler-Hadwiger inequality”, The Mathematical Gazette 106, issue 565, March 2022, pp. 138-143.

If you have The Geometer’s Sketchpad, you can view Gergonne point. If you have GeoGebra, you can view Gergonne point.

X(7) lies on the Lucas cubic and these lines: 1,20 2,9 3,943 4,273 6,294 8,65 11,658 12,1268 21,56 27,81 37,241 33,1041 34,1039 55,2346 58,272 59,1275 72,443 73,1246 76,1479 80,150 92,189 100,1004 104,934 108,1013 109,675 145,1266 171,983 174,234 177,555 190,344 192,335 193,239 218,277 220,1223 225,969 238,1471 253,280 256,982 274,959 281,653 286,331 310,314 330,1432 349,1269 354,479 404,1259 452,1467 464,1214 480,1376 492,1267 513,885 517,1000 528,664 554,1082 594,599 604,1429 757,1414 840,927 857,1901 870,1431 940,1407 941,1427 944,1389 952,1159 986,1254 987,1106 1002,1362 1020,1765 1061,1870 1210,3091 1354,1367 1365,1366 1386,1456 1419,1449 1435,1848 1486,1602 1617,1621 2475,2893

X(7) is the {X(69),:ref:X(75) <X(75)>}-harmonic conjugate of X(8). For a list of other harmonic conjugates of X(7), click Tables at the top of this page.

X(7) = reflection of X(i) in X(j) for these (i,j): (9,142), (144,9), (390,1), (673,1086), (1156,11)

X(7) = isogonal conjugate of X(55)

X(7) = isotomic conjugate of X(8)

X(7) = cyclocevian conjugate of X(7)

X(7) = circumcircle-inverse of (32624)

X(7) = incircle-inverse of (1323)

X(7) = complement of X(144)

X(7) = anticomplement of X(9)

X(7) = complementary conjugate of X(2884)

X(7) = anticomplementary conjugate of X(329)

X(7) = X(i)-Ceva conjugate of X(j) for these (i,j): (75,347), (85,2), (86,77), (286,273), (331,278)

X(7) = cevapoint of X(i) and X(j) for these (i,j): (1,57), (2,145), (4,196), (11,514), (55,218), (56,222), (63,224), (65,226), (81,229), (177,234)

X(7) = X(i)-cross conjugate of X(j) for these (i,j): (1,2), (4,189), (11,514), (55,277), (56,278), (57,279), (65,57), (177,174), (222,348), (226,85), (354,1), (497,8), (517,88)

X(7) = crosspoint of X(i) and X(j) for these (i,j): (75,309), (86,286)

X(7) = crosssum of X(i) and X(j) for these (i,j): (41,1253), (42,228)

X(7) = crossdifference of every pair of points on line X(657)X(663)

X(7) = X(57)-Hirst inverse of X(1447)

X(7) = insimilicenter of inner and outer Soddy circles; the exsimilicenter is X(1)

X(7) = X(i)-beth conjugate of X(j) for these (i,j): (7,269), (21,991), (75,75), (86,7), (99,7), (190,7), (290,7), (314,69), (645,344), (648,7), (662,6), (664,7), (666,7), (668,7), (670,7), (671,7), (811,264), (886,7), (889,7), (892,7), (903,7)

X(7) = vertex conjugate of foci of inellipse that is isotomic conjugate of isogonal conjugate of incircle (centered at X(2886))

X(7) = trilinear product of vertices of Hutson-extouch triangle

X(7) = orthocenter of X(4)X(8)X(885)

X(7) = trilinear cube of X(506)

X(7) = barycentric product of PU(47)

X(7) = trilinear product of PU(94)

X(7) = vertex conjugate of PU(95)

X(7) = bicentric sum of PU(120)

X(7) = perspector of ABC and its intouch triangle.

X(7) = perspector of ABC and the reflection in X(57) of the pedal triangle of X(57)

X(7) = perspector of AC-incircle

X(7) = X(6)-of-extraversion triangle-of-X(8)

X(7) = X(6)-of-intouch-triangle; X(7) is the only point X inside ABC Such that X(ABC) = X(A’B’C’), where A’B’C’ = cevian triangle of X

X(7) = {X(2),:ref:X(63) <X(63)>}-harmonic conjugate of X(5273)

X(7) = {X(9),:ref:X(57) <X(57)>}-harmonic conjugate of X(1445)

X(7) = {X(1371),:ref:X(1372) <X(1372)>}-harmonic conjugate of X(1)

X(7) = {X(1373),:ref:X(1374) <X(1374)>}-harmonic conjugate of X(1)

X(7) = trilinear pole of Gergonne line

X(7) = trilinear pole, wrt intouch triangle, of Gergonne line

X(7) = pole of Gergonne line wrt incircle

X(7) = pole wrt polar circle of trilinear polar of X(281) (line X(3064)X(3700))

X(7) = X(48)-isoconjugate (polar conjugate) of X(281)

X(7) = X(6)-isoconjugate of X(9)

X(7) = X(75)-isoconjugate of X(2175)

X(7) = X(1101)-isoconjugate of X(4092)

X(7) = perspector of circumconic centered at X(3160)

X(7) = center of circumconic that is locus of trilinear poles of lines passing through X(3160)

X(7) = X(2)-Ceva conjugate of X(3160)

X(7) = antigonal image of X(1156)

X(7) = homothetic center of intouch triangle and anticomplement of the excentral triangle

X(7) = X(6)-of-intouch-triangle; X(7) is the only point X inside ABC such that X(ABC) = X(A’B’C’), where A’B’C’ = cevian triangle of X

X(7) = perspector of ABC and cross-triangle of inner and outer Soddy triangles

X(7) = perspector of excentral triangle and cross-triangle of ABC and Honsberger triangle

X(7) = perspector of inverse-in-excircles triangle and cross-triangle of ABC and inverse-in-excircles triangle

X(7) = perspector of inverse-in-incircle triangle and cross-triangle of ABC and inverse-in-incircle triangle

X(7) = X(1843)-of-excentral-triangle

X(7) = Cundy-Parry Phi transform of X(943)

X(7) = Cundy-Parry Psi transform of X(942)

X(7) = {X(1),:ref:X(1742) <X(1742)>}-harmonic conjugate of X(2293)

X(7) = barycentric square of X(508)

X(7) = perspector of ABC and cross-triangle of ABC and Gemini triangle 40

X(7) = barycentric product of vertices of Gemini triangle 40

X(7) = excentral-to-intouch similarity image of X(9)

X(7) = circumconic-centered-at-X(9)-inverse of X(37787)

X(7) = endo-homothetic center of Ehrmann vertex-triangle and Ehrmann mid-triangle; the homothetic center is X(3818)