X2594
SUM OF PU(68)
The notation “PU(68)”, as described in the Bicentric Pairs section accessible by the Tables button, abbreviates a bicentric pair of points P =\(p : q : r\) and U =\(u : v : w\) , where p, q, r have the form g(a,b,c), g(b,c,a), g(c,a,b) and u,v,w have the form h(a,b,c), h(b,c,a), h(c,a,b). “Sum of PU(68)” denotes the point
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, 5 }: Feuerbach Line { 42, 65 }: Central Line L(42, 65) { 59, 60 }: Central Line L(59, 60) { 56, 181 }: Central Line L(56, 181) { 35, 500 }: Central Line L(35, 500) { 55, 581 }: Central Line L(55, 581)
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_2599 = SUM OF PU(69) (0.5631, 0.2411) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_59 = ISOGONAL CONJUGATE OF X(11) (0.5333, 0.4355) X_60 = ISOGONAL CONJUGATE OF X(12) (0.5790, 0.2258) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_181 = APOLLONIUS POINT (0.6163, 0.3093) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_500 = ORTHOCENTER OF THE INCENTRAL TRIANGLE (0.5577, 0.1973) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_581 = ORTHOHARMONIC OF X(284) (0.5657, 0.2106) X_2594 = SUM OF PU(68) (0.5791, 0.2254)
A
B
C
X1
X2599
X5
X42
X65
X59
X60
X56
X181
X35
X500
X55
X581
X2594
(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, 5 }: Feuerbach Line { 42, 65 }: Central Line L(42, 65) { 59, 60 }: Central Line L(59, 60) { 56, 181 }: Central Line L(56, 181) { 35, 500 }: Central Line L(35, 500) { 55, 581 }: Central Line L(55, 581)
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_2599 = SUM OF PU(69) (0.5804, 0.2196) X_5 = NINE-POINT CENTER (0.5700, 0.2400) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.6250, 0.2188) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6400, 0.2800) X_59 = ISOGONAL CONJUGATE OF X(11) (0.6000, 0.4000) X_60 = ISOGONAL CONJUGATE OF X(12) (0.6152, 0.1823) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6667, 0.3333) X_181 = APOLLONIUS POINT (0.6545, 0.2970) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5556, 0.1111) X_500 = ORTHOCENTER OF THE INCENTRAL TRIANGLE (0.5833, 0.1429) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5714, 0.1429) X_581 = ORTHOHARMONIC OF X(284) (0.5946, 0.1622) X_2594 = SUM OF PU(68) (0.6154, 0.1795)
A
B
C
X1
X2599
X5
X42
X65
X59
X60
X56
X181
X35
X500
X55
X581
X2594
(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, 5 }: Feuerbach Line { 42, 65 }: Central Line L(42, 65) { 59, 60 }: Central Line L(59, 60) { 56, 181 }: Central Line L(56, 181) { 35, 500 }: Central Line L(35, 500) { 55, 581 }: Central Line L(55, 581)
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_2599 = SUM OF PU(69) (0.5675, 0.2217) X_5 = NINE-POINT CENTER (0.5590, 0.2429) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.6052, 0.2210) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6180, 0.2838) X_59 = ISOGONAL CONJUGATE OF X(11) (0.5932, 0.4263) X_60 = ISOGONAL CONJUGATE OF X(12) (0.5969, 0.1840) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6411, 0.3392) X_181 = APOLLONIUS POINT (0.6305, 0.3013) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5465, 0.1118) X_500 = ORTHOCENTER OF THE INCENTRAL TRIANGLE (0.5699, 0.1439) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5599, 0.1439) X_581 = ORTHOHARMONIC OF X(284) (0.5794, 0.1635) X_2594 = SUM OF PU(68) (0.5970, 0.1810)
A
B
C
X1
X2599
X5
X42
X65
X59
X60
X56
X181
X35
X500
X55
X581
X2594
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, 5 }: Feuerbach Line { 42, 65 }: Central Line L(42, 65) { 59, 60 }: Central Line L(59, 60) { 56, 181 }: Central Line L(56, 181) { 35, 500 }: Central Line L(35, 500) { 55, 581 }: Central Line L(55, 581)
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_2599 = SUM OF PU(69) (-0.4936, 0.3080) X_5 = NINE-POINT CENTER (0.5666, 0.2774) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.6461, 0.1459) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6417, 0.2223) X_59 = ISOGONAL CONJUGATE OF X(11) (0.6302, 0.2712) X_60 = ISOGONAL CONJUGATE OF X(12) (0.6473, 0.0852) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6494, 0.2495) X_181 = APOLLONIUS POINT (0.6498, 0.2312) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5793, 0.0033) X_500 = ORTHOCENTER OF THE INCENTRAL TRIANGLE (0.6005, 0.0064) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5940, 0.0549) X_581 = ORTHOHARMONIC OF X(284) (0.6131, 0.0418) X_2594 = SUM OF PU(68) (0.6538, 0.0142)
A
B
C
X1
X2599
X5
X42
X65
X59
X60
X56
X181
X35
X500
X55
X581
X2594
Kimberling (6, 9, 13) | AB = 1
X2594 (SUM OF PU(68) ) 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 \(f(A,B,C) = \cos(C- A) + \cos(B - A)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(2.584476245300\)
▸ Incident Central Lines23 lines
{1 5 6 2197 35 500 42 65 55 581 56 181 59 60 319 1273 354 1066 523 2616 654 2598 692 3145 995 1388 1048 2595 1064 3057 1193 1319 1203 2078 1324 1437 1745 1836 1788 3240 2151 2307 2599 2611 2601 2603
▸ Functional Relations7 relations
Notes
g(a,b,c) + h(a,b,c) : g(b,c,a) + h(b,c,a) : g(c,a,b) + h(c,a,b),
where (P, Q) are the bicentric pair (P(68),Q(68)) as listed in the Bicentric Pairs section.
