X2534
INSIMILICENTER(INCIRCLE, SPIEKER RADICAL CIRCLE)
The Spieker radical circle, with center
X10 and radius d/2, is introduced in
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) { 1, 2 }: Nagel Line { 11, 2540 }: Central Line L(11, 2540) { 12, 2541 }: Central Line L(12, 2541) { 55, 2537 }: Central Line L(55, 2537) { 56, 2536 }: Central Line L(56, 2536) { 181, 2538 }: Central Line L(181, 2538)
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_2 = CENTROID (0.5374, 0.2000) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_2540 = INSIMILICENTER(NINE-POINT CIRCLE, SPIEKER RADICAL CIRCLE) (0.5461, 0.2305) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_2541 = EXSIMILICENTER(NINE-POINT CIRCLE, SPIEKER RADICAL CIRCLE) (0.5790, 0.2909) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_2537 = EXSIMILICENTER(CIRCUMCIRCLE, SPIEKER RADICAL CIRCLE) (0.4272, -0.1877) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_2536 = INSIMILICENTER(CIRCUMCIRCLE, SPIEKER RADICAL CIRCLE) (0.5089, 0.1377) X_181 = APOLLONIUS POINT (0.6163, 0.3093) X_2538 = INSIMILICENTER(APOLLONIUS CIRCLE, SPIEKER RADICAL CIRCLE) (0.4530, 0.0595) X_2534 = INSIMILICENTER(INCIRCLE, SPIEKER RADICAL CIRCLE) (0.5580, 0.2201)
A
B
C
X1
X2
X11
X2540
X12
X2541
X55
X2537
X56
X2536
X181
X2538
X2534
(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) { 1, 2 }: Nagel Line { 11, 2540 }: Central Line L(11, 2540) { 12, 2541 }: Central Line L(12, 2541) { 55, 2537 }: Central Line L(55, 2537) { 56, 2536 }: Central Line L(56, 2536) { 181, 2538 }: Central Line L(181, 2538)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_1 = INCENTER (0.6000, 0.2000) X_2 = CENTROID (0.5467, 0.1600) X_11 = FEUERBACH POINT (0.7200, 0.0400) X_2540 = INSIMILICENTER(NINE-POINT CIRCLE, SPIEKER RADICAL CIRCLE) (0.5554, 0.2109) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5867, 0.2178) X_2541 = EXSIMILICENTER(NINE-POINT CIRCLE, SPIEKER RADICAL CIRCLE) (0.6049, 0.3098) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5714, 0.1429) X_2537 = EXSIMILICENTER(CIRCUMCIRCLE, SPIEKER RADICAL CIRCLE) (0.4076, -0.6465) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6667, 0.3333) X_2536 = INSIMILICENTER(CIRCUMCIRCLE, SPIEKER RADICAL CIRCLE) (0.5090, 0.0632) X_181 = APOLLONIUS POINT (0.6545, 0.2970) X_2538 = INSIMILICENTER(APOLLONIUS CIRCLE, SPIEKER RADICAL CIRCLE) (0.4284, -0.0431) X_2534 = INSIMILICENTER(INCIRCLE, SPIEKER RADICAL CIRCLE) (0.5802, 0.1852)
A
B
C
X1
X2
X11
X2540
X12
X2541
X55
X2537
X56
X2536
X181
X2538
X2534
(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) { 1, 2 }: Nagel Line { 11, 2540 }: Central Line L(11, 2540) { 12, 2541 }: Central Line L(12, 2541) { 55, 2537 }: Central Line L(55, 2537) { 56, 2536 }: Central Line L(56, 2536) { 181, 2538 }: Central Line L(181, 2538)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_1 = INCENTER (0.5841, 0.2021) X_2 = CENTROID (0.5393, 0.1620) X_11 = FEUERBACH POINT (0.6897, 0.0298) X_2540 = INSIMILICENTER(NINE-POINT CIRCLE, SPIEKER RADICAL CIRCLE) (0.5468, 0.2135) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5729, 0.2203) X_2541 = EXSIMILICENTER(NINE-POINT CIRCLE, SPIEKER RADICAL CIRCLE) (0.5882, 0.3132) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5599, 0.1439) X_2537 = EXSIMILICENTER(CIRCUMCIRCLE, SPIEKER RADICAL CIRCLE) (0.4224, -0.6480) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6411, 0.3392) X_2536 = INSIMILICENTER(CIRCUMCIRCLE, SPIEKER RADICAL CIRCLE) (0.5077, 0.0639) X_181 = APOLLONIUS POINT (0.6305, 0.3013) X_2538 = INSIMILICENTER(APOLLONIUS CIRCLE, SPIEKER RADICAL CIRCLE) (0.4400, -0.0431) X_2534 = INSIMILICENTER(INCIRCLE, SPIEKER RADICAL CIRCLE) (0.5674, 0.1871)
A
B
C
X1
X2
X11
X2540
X12
X2541
X55
X2537
X56
X2536
X181
X2538
X2534
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) { 1, 2 }: Nagel Line { 11, 2540 }: Central Line L(11, 2540) { 12, 2541 }: Central Line L(12, 2541) { 55, 2537 }: Central Line L(55, 2537) { 56, 2536 }: Central Line L(56, 2536) { 181, 2538 }: Central Line L(181, 2538)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_1 = INCENTER (0.6154, 0.1300) X_2 = CENTROID (0.5444, 0.0934) X_11 = FEUERBACH POINT (0.6563, 0.0066) X_2540 = INSIMILICENTER(NINE-POINT CIRCLE, SPIEKER RADICAL CIRCLE) (0.5467, 0.2076) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.6001, 0.1762) X_2541 = EXSIMILICENTER(NINE-POINT CIRCLE, SPIEKER RADICAL CIRCLE) (0.6306, 0.5018) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5940, 0.0549) X_2537 = EXSIMILICENTER(CIRCUMCIRCLE, SPIEKER RADICAL CIRCLE) (0.6801, 6.8228) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6494, 0.2495) X_2536 = INSIMILICENTER(CIRCUMCIRCLE, SPIEKER RADICAL CIRCLE) (0.5045, -0.0954) X_181 = APOLLONIUS POINT (0.6498, 0.2312) X_2538 = INSIMILICENTER(APOLLONIUS CIRCLE, SPIEKER RADICAL CIRCLE) (0.4204, -0.2353) X_2534 = INSIMILICENTER(INCIRCLE, SPIEKER RADICAL CIRCLE) (0.5948, 0.1194)
A
B
C
X1
X2
X11
X2540
X12
X2541
X55
X2537
X56
X2536
X181
X2538
X2534
Kimberling (6, 9, 13) | AB = 1
X2534 (INSIMILICENTER(INCIRCLE, SPIEKER RADICAL CIRCLE) ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\) .
Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) , where \(f(a,b,c) = bc[(b + c)r + ad]\) , where \(d = (r^{2} + s^{2})^{1/2}\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(1.962693027102\)
▸ Incident Central Lines7 lines
{1 2 11 2540 12 2541 55 2537 56 2536 181 2538 1682 2539
Notes
Darij Grinberg and Paul Yiu, “The Apollonius Circle as a Tucker Circle,” Forum Geometricorum 2 (2002) 175-182.
