X3033
INTERSECTION X10 X_116∩X43 X_57
X3033 = inverse-in-excircles-radical-circle of
X120
5, 6, 7
3, 4, 5
Kepler
Kimberling
< 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) { 43, 57 }: Central Line L(43, 57) { 10, 116 }: Central Line L(10, 116) { 101, 386 }: Central Line L(101, 386) { 103, 573 }: Central Line L(103, 573) { 118, 2051 }: Central Line L(118, 2051) { 440, 2968 }: Central Line L(440, 2968)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_43 = -CEVA CONJUGATE OF X(1) (0.6358, 0.2963) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_116 = MIDPOINT OF X(4) AND X(103) (0.6990, 0.0375) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.2143, 0.5249) X_386 = INVERSE-IN-BROCARD-CIRCLE OF X(58) (0.5912, 0.2526) X_103 = ANTIPODE OF X(101) (0.7857, -0.3208) X_573 = ORTHOHARMONIC OF X(58) (0.4486, 0.0172) X_118 = MIDPOINT OF X(4) AND X(101) (0.4133, 0.4603) X_2051 = EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE) (0.6034, 0.3358) X_440 = COMPLEMENT OF X(27) (0.4970, 0.0943) X_2968 = 2nd MACBEATH POINT (0.5881, 0.0121) X_3033 = INTERSECTION X(10)X(116)∩X(43)X(57) (-0.7602, 1.2289)
A
B
C
X43
X57
X10
X116
X101
X386
X103
X573
X118
X2051
X440
X2968
X3033
(a, b, c) = (5, 6, 7) | AB = 1
< 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) { 43, 57 }: Central Line L(43, 57) { 10, 116 }: Central Line L(10, 116) { 101, 386 }: Central Line L(101, 386) { 103, 573 }: Central Line L(103, 573) { 118, 2051 }: Central Line L(118, 2051) { 1490, 2808 }: Central Line L(1490, 2808)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_43 = -CEVA CONJUGATE OF X(1) (0.6941, 0.2706) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6500, 0.3000) X_10 = SPIEKER CENTER (0.5200, 0.1400) X_116 = MIDPOINT OF X(4) AND X(103) (0.6662, 0.0092) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.3077, 0.4615) X_386 = INVERSE-IN-BROCARD-CIRCLE OF X(58) (0.6296, 0.2222) X_103 = ANTIPODE OF X(101) (0.6923, -0.4615) X_573 = ORTHOHARMONIC OF X(58) (0.4255, -0.1277) X_118 = MIDPOINT OF X(4) AND X(101) (0.4738, 0.4708) X_2051 = EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE) (0.6356, 0.3713) X_1490 = ∩X(3)X(9) (0.4000, -1.2000) X_2808 = ISOGONAL CONJUGATE OF X(2724) (0.3255, 0.6908) X_3033 = INTERSECTION X(10)X(116)∩X(43)X(57) (-0.5615, 1.1077)
A
B
C
X43
X57
X10
X116
X101
X386
X103
X573
X118
X2051
X1490
X2808
X3033
(a, b, c) = (3, 4, 5) | AB = 1
< 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) { 43, 57 }: Central Line L(43, 57) { 10, 116 }: Central Line L(10, 116) { 101, 386 }: Central Line L(101, 386) { 103, 573 }: Central Line L(103, 573) { 118, 2051 }: Central Line L(118, 2051) { 1490, 2808 }: Central Line L(1490, 2808)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_43 = -CEVA CONJUGATE OF X(1) (0.6641, 0.2739) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6266, 0.3045) X_10 = SPIEKER CENTER (0.5170, 0.1419) X_116 = MIDPOINT OF X(4) AND X(103) (0.6408, 0.0067) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.3365, 0.4725) X_386 = INVERSE-IN-BROCARD-CIRCLE OF X(58) (0.6091, 0.2246) X_103 = ANTIPODE OF X(101) (0.6635, -0.4725) X_573 = ORTHOHARMONIC OF X(58) (0.4375, -0.1285) X_118 = MIDPOINT OF X(4) AND X(101) (0.4773, 0.4792) X_2051 = EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE) (0.6143, 0.3758) X_1490 = ∩X(3)X(9) (0.4159, -1.2021) X_2808 = ISOGONAL CONJUGATE OF X(2724) (0.3513, 0.7053) X_3033 = INTERSECTION X(10)X(116)∩X(43)X(57) (-0.3931, 1.1360)
A
B
C
X43
X57
X10
X116
X101
X386
X103
X573
X118
X2051
X1490
X2808
X3033
Kepler (1, √φ, φ) | AB = 1
< 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) { 43, 57 }: Central Line L(43, 57) { 10, 116 }: Central Line L(10, 116) { 101, 386 }: Central Line L(101, 386) { 103, 573 }: Central Line L(103, 573) { 118, 2051 }: Central Line L(118, 2051) { 440, 2968 }: Central Line L(440, 2968)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_43 = -CEVA CONJUGATE OF X(1) (0.7237, 0.1860) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6451, 0.2341) X_10 = SPIEKER CENTER (0.5089, 0.0750) X_116 = MIDPOINT OF X(4) AND X(103) (0.5925, -0.0067) X_101 = Ψ(INCENTER, SYMMEDIAN POINT) (0.4482, 0.2934) X_386 = INVERSE-IN-BROCARD-CIRCLE OF X(58) (0.6513, 0.1486) X_103 = ANTIPODE OF X(101) (0.5518, -0.8428) X_573 = ORTHOHARMONIC OF X(58) (0.4096, -0.5275) X_118 = MIDPOINT OF X(4) AND X(101) (0.5407, 0.5614) X_2051 = EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE) (0.6409, 0.5381) X_440 = COMPLEMENT OF X(27) (0.5165, -0.1382) X_2968 = 2nd MACBEATH POINT (0.4091, -0.0004) X_3033 = INTERSECTION X(10)X(116)∩X(43)X(57) (-0.1542, 0.7231)
A
B
C
X43
X57
X10
X116
X101
X386
X103
X573
X118
X2051
X440
X2968
X3033
Kimberling (6, 9, 13) | AB = 1
X3033 (INTERSECTION X10 X_116∩X43 X_57 ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(a[(a^{2} - bc)^{2}(b + c)^{2} - (b^{3} - c^{3})^{2}] : :\)
Search Value (6, 9, 13): \(1.632334637245\)
X3033
▸ Incident Central Lines9 lines
{10 116 43 57 101 386 103 573 118 2051 440 2968 1490 2808 2774 3031 2786 3029
▸ Functional Relations2 relations
Notes
Let A' be the inverse-in-excircles-radical-circle of the midpoint of BC , and define B and C' cyclically. Triangle A'B'C' is inscribed in the Apollonius circle and homothetic to the excentral triangle . X3032 X98 A'B'C' . (Randy Hutson , August 29, 2018)
