X₁

INCENTER

incenter(A, B, C) -> X_1 • Euclid IV.4 • Kimberling X(1) • Search Value \((6, 9, 13)\): \(\frac{2 \sqrt{35}}{7} \approx 1.690308509457\)

X₁ is the point of concurrence of the interior angle bisectors of ABC; the point inside ABC whose distances from sidelines BC, CA, AB are equal. This equal distance, r, is the radius of the incircle, given by \(r = 2 \operatorname{area}(ABC)/(a + b + c)\).

15, 13, 14 3, 4, 5 Kepler Kimberling

X₁ (INCENTER) constructed in exact rational coordinates on the Heronian reference triangle \((a, b, c) = (15, 13, 14)\) over \(\mathbb{Q}\) with vertices A = \((0, 0)\), B = \((14, 0)\), C = \((5, 12)\), area \(\Delta = 84\), and inradius \(r = 4\). Switch tabs above to inspect exact constructions on Right \((3, 4, 5)\), Kepler \((1, \sqrt{\phi}, \phi)\), and Kimberling \((6, 9, 13)\).

Constructed Elements

• Auxiliary Points: I_a, I_b, I_c — Internal angle bisector feet on BC, CA, AB

• Auxiliary Points: T_a, T_b, T_c — Incircle touchpoints (right-spread pedal feet) on BC, CA, AB

• Circle: Incircle (incircle) — Inscribed circle tangent to sidelines at T_a, T_b, T_c; centered at X₁ INCENTER

• Concurrence Lines: A–I_a, B–I_b, C–I_c concurring at X₁

Homogeneous Coordinates & Exact Invariants

Homogeneous Coordinates of X₁ on Heronian \((15, 13, 14)\): orthogonal trilinear struts \((\alpha, \beta, \gamma)\) and shaded barycentric sub-triangles \((\Delta_a, \Delta_b, \Delta_c)\).

• Trilinear Coordinates (\(\alpha : \beta : \gamma = 1 : 1 : 1\)): Dropping orthogonal perpendiculars X₁T_a, X₁T_b, and X₁T_c from X₁ to sidelines BC, CA, and AB meets the sides at right-spread touchpoints T_a = \(\left(\frac{46}{5}, \frac{32}{5}\right)\), T_b = \(\left(\frac{30}{13}, \frac{72}{13}\right)\), and T_c = \((6, 0)\). Every strut equals the exact inradius \(r = 4\) (quadrance \(Q_r = 16\)), realizing \(4 : 4 : 4 = 1 : 1 : 1\).

• Barycentric Coordinates (\(u : v : w = a : b : c\)): Joining X₁ to the vertices along X₁A, X₁B, and X₁C partitions \(\triangle ABC\) into three sub-triangles △X₁BC, △X₁CA, and △X₁AB. Sharing common altitude \(r = 4\), their exact signed areas on \((15, 13, 14)\) are \(\Delta_a = \frac{1}{2}(15)(4) = 30\), \(\Delta_b = \frac{1}{2}(13)(4) = 26\), and \(\Delta_c = \frac{1}{2}(14)(4) = 28\) (\(30 + 26 + 28 = 84 = \Delta\)), in exact ratio \(30 : 26 : 28 = 15 : 13 : 14 = a : b : c\).

Exact Symbolic Coordinates & Invariants Across the Four Reference Triangles

Reference Triangle	Field	Cartesian \((x, y)\)	Trilinears \(\alpha : \beta : \gamma\)	Barycentrics \(u : v : w\)	Sub-Areas \((\Delta_a, \Delta_b, \Delta_c)\)	Inradius & Quadrance \((r,\, Q_r)\)
Heronian \((15, 13, 14)\)	\(\mathbb{Q}\)	\(\left(6,\; 4\right)\)	\(1 : 1 : 1\)	\(15 : 13 : 14\)	\(\left(30,\; 26,\; 28\right)\)	\(r = 4,\; Q_r = 16\)
Right \((3, 4, 5)\)	\(\mathbb{Q}\)	\(\left(3,\; 1\right)\)	\(1 : 1 : 1\)	\(3 : 4 : 5\)	\(\left(\frac{3}{2},\; 2,\; \frac{5}{2}\right)\)	\(r = 1,\; Q_r = 1\)
Kepler \((1, \sqrt{\phi}, \phi)\)	\(\mathbb{Q}(\sqrt{\phi})\)	\(\left(\frac{\sqrt{\phi}}{2} + \frac{\phi}{2} - \frac{1}{2},\; \frac{\sqrt{\phi}}{2} - \frac{\phi}{2} + \frac{1}{2}\right)\)	\(1 : 1 : 1\)	\(1 : \sqrt{\phi} : \phi\)	\(\left(\frac{\sqrt{\phi}}{4} - \frac{\phi}{4} + \frac{1}{4},\; - \frac{\phi \sqrt{\phi}}{4} + \frac{\sqrt{\phi}}{4} + \frac{\phi}{4},\; \frac{\phi \sqrt{\phi}}{4} - \frac{1}{4}\right)\)	\(r = \frac{\sqrt{\phi}}{2} - \frac{\phi}{2} + \frac{1}{2},\; Q_r = - \frac{\phi \sqrt{\phi}}{2} + \frac{\sqrt{\phi}}{2} + \frac{1}{2}\)
Kimberling \((6, 9, 13)\)	\(\mathbb{Q}(\sqrt{35})\)	\(\left(8,\; \frac{2 \sqrt{35}}{7}\right)\)	\(1 : 1 : 1\)	\(6 : 9 : 13\)	\(\left(\frac{6 \sqrt{35}}{7},\; \frac{9 \sqrt{35}}{7},\; \frac{13 \sqrt{35}}{7}\right)\)	\(r = \frac{2 \sqrt{35}}{7},\; Q_r = \frac{20}{7}\)

Trilinears: \(1 : 1 : 1\)

Barycentrics: \(a : b : c\)

Tripolars: \(\sqrt{b c (b + c - a)} : :\)

Search Value (6, 9, 13): \(\frac{2 \sqrt{35}}{7} \approx 1.690308509457\)

▸ Alternate Coordinates2 forms

• Barycentrics: \(\sin A : \sin B : \sin C\)

• Tripolars: \(\sec A' : :\), where J_aJ_bJ_c is the excentral triangle

▸ Linear Combinations5 combinations

• X₁ = \(3R\) X₂ + \(r\) X₃ + \(s \cot(\omega )\) X₆

• X₁ = A / \(R_{a}\) + B / \(R_{b}\) + C / \(R_{c}\) - X₁₇₆ / \(R_{s}\), where \(R_{a}\), \(R_{b}\), \(R_{c}\) = radii of Soddy circles, \(R_{s}\) = radius of inner Soddy circle, A, B, C are the vertices of ABC

• X₁ = A / \(R_{a}\) + B / \(R_{b}\) + C / \(R_{c}\) - X₁₇₅ / \(R_{s}'\), where \(R_{a}\), \(R_{b}\), \(R_{c}\) = radii of Soddy circles, \(R_{s}'\) = radius of outer Soddy circle, A, B, C are the vertices of ABC

• X₁ = \((\sin A)\) A + \((\sin B)\) B + \((\sin C)\) C, where A, B, C are vertices of ABC

• X₁ = \(a\) A + \(b\) B + \(c\) C, where A, B, C are vertices

Incident Central Lines

Principal Central Lines Through X₁ on Heronian \((15, 13, 14)\).

• Nagel Line {1, 2, 8, 10} — collinear axis of X₁, X₂ (Centroid, \((\frac{19}{3}, 4)\)), X₁₀ (Spieker Center, \((\frac{13}{2}, 4)\)), and X₈ (Nagel Point, \((7, 4)\)). On \((15, 13, 14)\), its exact equation is \(y - 4 = 0\) (spread \(0\), parallel to baseline AB).

• Feuerbach Line {1, 5, 11} — collinear axis of X₁, X₅ (Nine-Point Center, \((6, \frac{63}{16})\)), and X₁₁ (Feuerbach Point, \((6, 8)\)). On \((15, 13, 14)\), its exact equation is \(x - 6 = 0\) (spread \(1\), perpendicular to baseline AB and orthogonal to the Nagel line).

• OI Line (Incenter–Circumcenter Axis) {1, 3, 40} — joins X₁, X₃ (Circumcenter, \((7, \frac{33}{8})\)), and X₄₀ (Bevan Point, \((8, \frac{17}{4})\)), with exact equation \(x - 8y + 26 = 0\) (spread \(\frac{1}{65}\)).

• Incenter–Symmedian Line {1, 6, 9} — joins X₁, X₆ (Symmedian Point, \((\frac{1673}{295}, \frac{1176}{295})\)), and X₉ (Mittenpunkt, \((\frac{973}{146}, \frac{294}{73})\)), with exact equation \(4x - 97y + 364 = 0\) (spread \(\frac{16}{9425}\)).

• Soddy Line {1, 7, 20} — joins X₁, X₇ (Gergonne Point, \((\frac{414}{73}, \frac{288}{73})\)), and X₂₀ (de Longchamps Point, \((9, \frac{9}{2})\)), with exact equation \(x - 6y + 18 = 0\) (spread \(\frac{1}{37}\)).

X₁ lies on all Z-cubics (e.g., Thomson, Darboux, Napoleon, Neuberg).

▸ Full Incident Central Line Catalog170 lines

{2, 8}, {3, 35}, {4, 33}, {5, 11}, {6, 9}, {7, 20}, {15, 1251}, {16, 1250}, {19, 28}, {21, 31}, {24, 1061}, {25, 1036}, {29, 92}, {30, 79}, {32, 172}, {39, 291}, {41, 101}, {49, 215}, {54, 3460}, {60, 110}, {61, 203}, {62, 202}, {64, 1439}, {69, 1245}, {71, 579}, {74, 3464}, {75, 86}, {76, 350}, {82, 560}, {84, 221}, {87, 192}, {88, 100}, {90, 155}, {99, 741}, {102, 108}, {104, 109}, {142, 277}, {147, 150}, {159, 1486}, {163, 293}, {164, 258}, {166, 1488}, {167, 174}, {168, 173}, {179, 1142}, {181, 970}, {182, 983}, {184, 1726}, {185, 296}, {188, 361}, {190, 537}, {195, 3467}, {196, 207}, {201, 212}, {204, 1712}, {210, 2334}, {224, 377}, {227, 1465}, {228, 1730}, {229, 267}, {256, 511}, {257, 385}, {280, 1256}, {281, 282}, {289, 363}, {312, 1089}, {318, 1897}, {320, 752}, {321, 964}, {329, 452}, {335, 384}, {336, 811}, {341, 1050}, {344, 1265}, {346, 1219}, {357, 1508}, {358, 1507}, {364, 365}, {371, 1702}, {372, 1703}, {376, 553}, {378, 1063}, {393, 836}, {394, 1711}, {399, 3065}, {409, 1247}, {410, 1248}, {411, 1254}, {442, 1834}, {474, 1339}, {475, 1861}, {512, 875}, {513, 764}, {514, 663}, {522, 1459}, {528, 1086}, {561, 718}, {563, 1820}, {564, 1048}, {572, 604}, {573, 941}, {574, 1571}, {594, 1224}, {607, 949}, {615, 3300}, {631, 1000}, {644, 1280}, {647, 1021}, {650, 1643}, {651, 1156}, {659, 891}, {662, 897}, {672, 1002}, {689, 719}, {704, 1502}, {727, 932}, {731, 789}, {748, 756}, {761, 825}, {765, 1052}, {810, 1577}, {840, 1308}, {905, 1734}, {908, 998}, {921, 1800}, {939, 1260}, {945, 1875}, {947, 1753}, {951, 1435}, {969, 1444}, {971, 1419}, {989, 1397}, {1013, 1430}, {1037, 1041}, {1053, 1110}, {1057, 1598}, {1059, 1597}, {1073, 3341}, {1075, 1148}, {1106, 1476}, {1157, 3483}, {1168, 1318}, {1170, 1253}, {1185, 1206}, {1197, 1613}, {1292, 1477}, {1333, 1761}, {1342, 1700}, {1343, 1701}, {1361, 1364}, {1389, 1393}, {1399, 1727}, {1406, 1480}, {1409, 1765}, {1437, 1710}, {1472, 1791}, {1719, 1790}, {1855, 1886}, {1859, 1871}, {1872, 1887}, {2120, 3461}, {2130, 3347}, {3183, 3345}, {3342, 3343}, {3344, 3351}, {3346, 3353}, {3348, 3472}, {3350, 3352}, {3354, 3355}, {3462, 3469}

Functional Relations & Conjugates

Conjugate & Homothetic Partners of X₁ on Heronian \((15, 13, 14)\).

• Self-Isogonal Fixed Point: X₁ is its own isogonal conjugate (\(1^{-1} : 1^{-1} : 1^{-1} = 1 : 1 : 1\)), acting as the identity element of trilinear multiplication.

• Isotomic Conjugate: X₇₅ (X₇₅, homogeneous barycentrics \(bc : ca : ab = 182 : 210 : 195\)) at exact rational coordinates \(\left(\frac{3915}{587}, \frac{2340}{587}\right)\), joined along {1, 75}.

• Anticomplement of Spieker Center: X₁ is the anticomplement of X₁₀ (Spieker Center, \(\left(\frac{13}{2}, 4\right)\)), dilated from homothetic center X₂ (Centroid, \(\left(\frac{19}{3}, 4\right)\)) by factor \(-2\) along {1, 10}.

• Complement of Nagel Point: X₁ is the complement of X₈ (Nagel Point, \((7, 4)\)), dilated from X₂ by factor \(-\frac{1}{2}\) along {1, 8}; equivalently, X₁₀ is the exact midpoint of X₁ and X₈.

• Circumcircle Inverse: Inversion in circumcircle centered at X₃ = \(\left(7, \frac{33}{8}\right)\) (\(R^2 = \frac{4225}{64}\)) maps X₁ along the OI axis (\(Q(X₁, X₃) = \frac{65}{64}\), scale \(R^2 / Q = 65\)) to X₃₆ at exact integer coordinates \((-58, -4)\).

▸ Full Functional Relations Catalog123 relations

• X₁ = midpoint of X_i and X_j for these (i,j): (3, 1482), (7, 390), (8, 145), (55, 2099), (56, 2098)

• X₁ = reflection of X_i in X_j for these (i,j): (2, 551), (3, 1385), (4, 946), (6, 1386), (8, 10), (9, 1001), (10, 1125), (11, 1387), (36, 1319), (40, 3), (43, 995), (46, 56), (57, 999), (63, 993), (65, 942), (72, 960), (80, 11), (100, 214), (191, 21), (200, 997), (238, 1279), (267, 229), (291, 1015), (355, 5), (484, 36), (984, 37), (1046, 58), (1054, 106), (1478, 226)

• X₁ = isogonal conjugate of X₁

• X₁ = isotomic conjugate of X₇₅

• X₁ = cyclocevian conjugate of X₁₀₂₉

• X₁ = circumcircle-inverse of X₃₆

• X₁ = Fuhrmann-circle-inverse of X₈₀

• X₁ = Bevan-circle-inverse of X₄₈₄

• X₁ = Spieker-radical-circle-inverse of X₃₈₄₇₁

• X₁ = complement of X₈

• X₁ = anticomplement of X₁₀

• X₁ = anticomplementary conjugate of X₁₃₃₀

• X₁ = complementary conjugate at X₁₃₂₉

• X₁ = eigencenter of cevian triangle of X_i for I = 1, 88, 100, 162, 190

• X₁ = eigencenter of anticevian triangle of X_i for I = 1, 44, 513

• X₁ = exsimilicenter of inner and outer Soddy circles; insimilicenter is X₇

• X₁ = X_i-Ceva conjugate of X_j for these (i,j): (2, 9), (4, 46), (6, 43), (7, 57), (8, 40), (9, 165), (10, 191), (21, 3), (29, 4), (75, 63), (77, 223), (78, 1490), (80, 484), (81, 6), (82, 31), (85, 169), (86, 2), (88, 44), (92, 19), (100, 513), (104, 36), (105, 238), (174, 173), (188, 164), (220, 170), (259, 503), (266, 361), (280, 84), (366, 364), (508, 362), (1492, 1491)

• X₁ = cevapoint of X_i and X_j for these (i,j): (2, 192), (6, 55), (11, 523), (15, 202), (16, 203), (19, 204), (31, 48), (34, 207), (37, 42), (50, 215), (56, 221), (65, 73), (244, 513)

• X₁ = X_i-cross conjugate of X_j for these (i,j): (2, 87), (3, 90), (6, 57), (31, 19), (33, 282), (37, 2), (38, 75), (42, 6), (44, 88), (48, 63), (55, 9), (56, 84), (58, 267), (65, 4), (73, 3), (192, 43), (207, 1490), (221, 40), (244, 513), (259, 258), (266, 505), (354, 7), (367, 366), (500, 35), (513, 100), (517, 80), (518, 291), (1491, 1492)

• X₁ = crosspoint of X_i and X_j for these (i,j): (2, 7), (8, 280), (21, 29), (59, 110), (75, 92), (81, 86)

• X₁ = crosssum of X_i and X_j for these (i,j): (2, 192), (4, 1148), (6, 55), (11, 523), (15, 202), (16, 203), (19, 204), (31, 48), (34, 207), (37, 42), (44, 678), (50, 215), (56, 221), (57, 1419), (65, 73), (214, 758), (244, 513), (500, 942), (512, 1015), (774, 820), (999, 1480)

• X₁ = crossdifference of every pair of points on line X₄₄X₅₁₃

• X₁ = X_i-Hirst inverse of X_j for these (i,j): (2, 239), (4, 243), (6, 238), (9, 518), (19, 240), (57, 241), (105, 294), (291, 292)

• X₁ = X₆-line conjugate of X₄₄

• X₁ = X_i-aleph conjugate of X_j for these (i,j): (1, 1), (2, 63), (4, 920), (21, 411), (29, 412), (88, 88), (100, 100), (162, 162), (174, 57), (188, 40), (190, 190), (266, 978), (365, 43), (366, 9), (507, 173), (508, 169), (513, 1052), (651, 651), (653, 653), (655, 655), (658, 658), (660, 660), (662, 662), (673, 673), (771, 771), (799, 799), (823, 823), (897, 897)

• X₁ = X_i-beth conjugate of X_j for these (i,j): (1, 56), (2, 948), (8, 8), (9, 45), (21, 1), (29, 34), (55, 869), (99, 85), (100, 1), (110, 603), (162, 208), (643, 1), (644, 1), (663, 875), (664, 1), (1043, 78)

• X₁ = insimilicenter of 1st & 2nd Johnson-Yff circles (the exsimilicenter is X₄)

• X₁ = orthic-isogonal conjugate of X₄₆

• X₁ = excentral-isogonal conjugate of X₄₀

• X₁ = excentral-isotomic conjugate of X₂₉₅₁

• X₁ = center of Conway circle

• X₁ = center of Adams circle

• X₁ = X₃ of polar triangle of Conway circle

• X₁ = homothetic center of intangents triangle and reflection of extangents triangle in X₃

• X₁ = Hofstadter 1/2 point

• X₁ = orthocenter of X₄X₉X₈₈₅

• X₁ = intersection of tangents at X₇ and X₈ to Lucas cubic K007

• X₁ = trilinear product of vertices of 2nd mixtilinear triangle

• X₁ = trilinear product of vertices of 2nd Sharygin triangle

• X₁ = homothetic center of Mandart-incircle triangle and 2nd isogonal triangle of X₁; see X₃₆

• X₁ = trilinear pole of the antiorthic axis (which is also the Monge line of the mixtilinear excircles)

• X₁ = pole wrt polar circle of trilinear polar of X₉₂ (line X₂₄₀X₅₂₂)

• X₁ = X₄₈-isoconjugate (polar conjugate) of X₉₂

• X₁ = X₆-isoconjugate of X₂

• X₁ = trilinear product of PU(i) for these i: 1, 17, 114, 115, 118, 119, 113

• X₁ = barycentric product of PU(i) for these i: 6, 124

• X₁ = vertex conjugate of PU(9)

• X₁ = bicentric sum of PU(i) for these i: 28, 47, 51, 55, 64

• X₁ = trilinear pole of PU(i) for these i: 33, 50, 57, 58, 74, 76, 78

• X₁ = crossdifference of PU(i) for these i: 33, 50, 57, 58, 74, 76, 78

• X₁ = midpoint of PU(i) for these i: 47, 51, 55

• X₁ = PU(28)-harmonic conjugate of X₁₀₂₃

• X₁ = PU(64)-harmonic conjugate of X₃₅₁

• X₁ = intersection of diagonals of trapezoid PU(6)PU(31)

• X₁ = perspector circumconic centered at X₉

• X₁ = eigencenter of mixtilinear excentral triangle

• X₁ = eigencenter of 2nd Sharygin triangle

• X₁ = perspector of ABC and unary cofactor triangle of extangents triangle

• X₁ = perspector of ABC and unary cofactor triangle of Feuerbach triangle

• X₁ = perspector of ABC and unary cofactor triangle of Apollonius triangle

• X₁ = perspector of ABC and unary cofactor triangle of 2nd mixtilinear triangle

• X₁ = perspector of ABC and unary cofactor triangle of 4th mixtilinear triangle

• X₁ = perspector of ABC and unary cofactor triangle of Apus triangle

• X₁ = perspector of unary cofactor triangles of 6th and 7th mixtilinear triangles

• X₁ = perspector of unary cofactor triangles of 2nd and 3rd extouch triangles

• X₁ = perspector of 5th mixtilinear triangle and unary cofactor triangle of 2nd mixtilinear triangle

• X₁ = perspector of 5th mixtilinear triangle and unary cofactor triangle of 4th mixtilinear triangle

• X₁ = X₃-of-reflection-triangle-of-X₁

• X₁ = X₁₁₈₁-of-2nd-extouch triangle

• X₁ = perspector of ABC and orthic-triangle-of-2nd-circumperp-triangle

• X₁ = X₄-of-excentral triangle

• X₁ = X₄₀-of-Yff central triangle

• X₁ = X₂₀-of-1st circumperp triangle

• X₁ = X₄-of-2nd circumperp triangle

• X₁ = X₄-of-Fuhrmann triangle

• X₁ = X₁₀₀-of-X₁-Brocard triangle

• X₁ = antigonal image of X₈₀

• X₁ = trilinear pole wrt excentral triangle of antiorthic axis

• X₁ = trilinear pole wrt incentral triangle of antiorthic axis

• X₁ = Miquel associate of X₇

• X₁ = homothetic center of Johnson triangle and cross-triangle of ABC and 2nd isogonal triangle of X₁

• X₁ = homothetic center of 1st Johnson-Yff triangle and cross-triangle of ABC and 2nd isogonal triangle of X₁

• X₁ = homothetic center of 2nd Johnson-Yff triangle and cross-triangle of ABC and 2nd isogonal triangle of X₁

• X₁ = homothetic center of Mandart-incircle triangle and cross-triangle of ABC and 1st Johnson-Yff triangle

• X₁ = homothetic center of medial triangle and cross-triangle of Aquila and anti-Aquila triangles

• X₁ = homothetic center of outer Garcia triangle and cross-triangle of Aquila and anti-Aquila triangles

• X₁ = X₈-of-cross-triangle-of Aquila-and-anti-Aquila-triangles

• X₁ = X₃-of-Mandart-incircle-triangle

• X₁ = X₁₀₀-of-inner-Garcia-triangle

• X₁ = Thomson-isogonal conjugate of X₁₆₅

• X₁ = X₈-of-outer-Garcia-triangle

• X₁ = X₄₈₆-of-BCI-triangle

• X₁ = X₁₆₄-of-orthic-triangle if ABC is acute

• X₁ = X₁₅₉₃-of-Ascella-triangle

• X₁ = excentral-to-Ascella similarity image of X₁₆₉₇

• X₁ = Dao image of X₁

• X₁ = X₄₀-of-reflection of ABC in X₃

• X₁ = radical center of the tangent circles of ABC

• X₁ = homothetic center of intangents triangle and anti-tangential midarc triangle

• X₁ = K(X₁₅) = K(X₁₆), as defined at X₁₇₄

• X₁ = X₃-of-hexyl-triangle

• X₁ = eigencenter of trilinear obverse triangle of X₂

• X₁ = hexyl-isogonal conjugate of X₄₀

• X₁ = inverse-in-polar-circle of X₁₇₈₅

• X₁ = inverse-in-orthoptic-circle-of-Steiner-inellipse of X₅₁₂₁

• X₁ = inverse-in-OI-inverter of X₁₁₅₅

• X₁ = inverse-in-Steiner-circumellipse of X₂₃₉

• X₁ = inverse-in-MacBeath-circumconic of X₂₃₂₃

• X₁ = inverse-in-circumconic-centered-at-X₉ of X₄₄

• X₁ = excentral-to-ABC barycentric image of X₄₀

• X₁ = excentral-to-ABC functional image of X₁₆₄

• X₁ = excentral-to-ABC trilinear image of X₁₆₄

• X₁ = orthic-to-ABC functional image of X₄, if ABC is acute

• X₁ = orthic-to-ABC trilinear image of X₄, if ABC is acute

• X₁ = intouch-to-ABC barycentric image of X₁

• X₁ = excentral-to-intouch similarity image of X₄₀

• X₁ = ABC-to-excentral barycentric image of X₈

• X₁ = X₁-vertex conjugate of X₅₆

• X₁ = perspector of ABC and reflection triangle of intangents triangle

• X₁ = perspector of pedal and anticevian triangles of X₄₀

• X₁ = perspector of hexyl triangle and antipedal triangle of X₄₀

• X₁ = perspector of hexyl triangle and anticevian triangle of X₅₇

• X₁ = X₄-of-Pelletier-triangle

Four-Triangle Symbolic Invariance

Four-Triangle Invariance of X₁ across Heronian (15, 13, 14), Right (3, 4, 5), Kepler (1, √φ, φ), and Kimberling (6, 9, 13).

Across all four number fields — rational \(\mathbb{Q}\) on Heronian (15, 13, 14) (\(X₁ = (6, 4)\), r = 4) and Pythagorean Right (3, 4, 5) (\(X₁ = (3, 1)\), r = 1), golden quadratic tower \(\mathbb{Q}(\sqrt{\phi})\) on Kepler (1, √φ, φ), and quadratic \(\mathbb{Q}(\sqrt{35})\) on Kimberling (6, 9, 13) (r = 2\sqrt{35}/7) — X₁ remains strictly interior, equidistant from all three sidelines (\(1 : 1 : 1\)), and partitions the triangle area in exact proportion \(a : b : c\).

Related Collection 1 Topic Guides

T_1: Isogonal Conjugate • T_2: Isotomic Conjugate • T_3: Complement & Anticomplement • T_4: Harmonic Conjugate & Cross-Ratio • T_5: Circle Inversion • T_7: The Nagel Line • T_9: The Feuerbach Line • T_10: The OI Line (Incenter–Circumcenter Axis) • T_11: The Soddy Line • T_12: The Fuhrmann & Spieker Central Lines • T_14: The Incircle & Excircles • T_16: The Nine-Point (Feuerbach) Circle • T_17: The Spieker Circle • T_20: Medial & Anticomplementary Triangles • T_22: Intouch (Contact) & Extouch Triangles • T_23: Incentral & Excentral Triangles

Notes

Three more points are also equidistant from the sidelines; they are given by these names and trilinears:

A-excenter = \(-1 : 1 : 1\), B-excenter =\(1 : -1 : 1\), C-excenter =\(1 : 1 : -1\).

The radii of the excircles are \(2 \operatorname{area}(ABC)/(-a + b + c)\), \(2 \operatorname{area}(ABC)/(a - b + c)\), \(2 \operatorname{area}(ABC)/(a + b - c)\).

Writing the A-exradius as r(a,b,c), the others are r(b,c,a) and r(c,a,b). If these exradii are abbreviated as \(r_{a}\), \(r_{b}\), \(r_{c}\), then \(1/r = 1/r_{a} +1/r_{b} + 1/r_{c}\). Moreover,

\(\operatorname{area}(ABC) = \sqrt{r r_{a} r_{b} r_{c}}\) and \(r_{a} + r_{b} + r_{c} = r + 4R\), where R denotes the radius of the circumcircle.

The incenter is the identity of the group of triangle centers under trilinear multiplication defined by \((x : y : z) \cdot (u : v : w) = xu : yv : zw\).

A construction for * is easily obtained from the construction for “barycentric multiplication” mentioned in connection with X₂, just below.

The incenter and the other classical centers are discussed in these highly recommended books:

Nathan Altshiller Court, College Geometry, Barnes & Noble, 1952;

• Roger A. Johnson, Advanced Euclidean Geometry, Dover, 1960.

Let O_A be the circle tangent to side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define O_B and O_C cyclically. Let L_A be the external tangent to circles O_B and O_C that is nearest to O_A. Define L_B and L_C cyclically. Let A' = L_B ∩L_C, and define B' and C' cyclically. Then triangle A'B'C' is homothetic to the medial triangle, and the center of homothety is X₁; see the reference at X₁₀₀₁.

Let A'B'C' and A″B″C″ be the intouch and excentral triangles; X₁ is the radical center of the circumcircles of AA'A″, BB'B″, CC'C″. (Randy Hutson, December 10, 2016)

Let Eu_aEu_bEu_c be the Feuerbach triangle. Let A″ be the trilinear pole of line Eu_bEu_c, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁. (Randy Hutson, November 17, 2019)

Let A'B'C' be the mixtilinear excentral triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁. (Randy Hutson, November 17, 2019)

Let A'B'C' be the mixtilinear excentral triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁. (Randy Hutson, November 17, 2019)

Let A'B'C' be the mixtilinear excentral triangle. Let L_A be the trilinear polar of A', and define L_B and L_C cyclically. Let A″ = L_B∩L_C, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₁. (Randy Hutson, November 17, 2019)

Let O_A be the circle centered at the A-vertex of the excenters-midpoints triangle and passing through A; define O_B and O_C cyclically. X₁ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

Let O_A be the circle centered at the A-vertex of the Gemini triangle 22 and passing through A; define O_B and O_C cyclically. X₁ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

X₁ is the {X₂,X₈}-harmonic conjugate of X₁₀.