X3082
RADICAL CENTER OF EXTERNAL MALFATTI CIRCLES
Let (
OA ) be the I-excircle of triangle
IBC , where I =
X1 , and define (
OB ) and (
OC ) cyclically. Let
A' be the touchpoint of (
OA ) and side
BC , and define
B' and
C' cyclically. Let
A″ be the insimilicenter of (
OB ) and (
OC ), and define
B″ and
C″ cyclically. The lines
A'A″ ,
B'B″ ,
C'C″ concur in
X3082 . (
Randy Hutson , July 11, 2019)
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) { 8, 178 }: Central Line L(8, 178) { 174, 176 }: Central Line L(174, 176)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_8 = NAGEL POINT (0.4694, 0.1333) X_178 = 2nd MID-ARC POINT (0.5439, 0.2098) X_174 = YFF CENTER OF CONGRUENCE (0.5813, 0.2549) X_176 = EQUAL DETOUR POINT (0.5803, 0.2537) X_3082 = RADICAL CENTER OF EXTERNAL MALFATTI CIRCLES (0.4624, 0.1261)
A
B
C
X8
X178
X174
X176
X3082
(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) { 8, 178 }: Central Line L(8, 178) { 174, 176 }: Central Line L(174, 176)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) X_8 = NAGEL POINT (0.4400, 0.0800) X_178 = 2nd MID-ARC POINT (0.5559, 0.1735) X_174 = YFF CENTER OF CONGRUENCE (0.6119, 0.2308) X_176 = EQUAL DETOUR POINT (0.6104, 0.2296) X_3082 = RADICAL CENTER OF EXTERNAL MALFATTI CIRCLES (0.4304, 0.0722)
A
B
C
X8
X178
X174
X176
X3082
(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) { 8, 178 }: Central Line L(8, 178) { 174, 176 }: Central Line L(174, 176)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) X_8 = NAGEL POINT (0.4499, 0.0817) X_178 = 2nd MID-ARC POINT (0.5471, 0.1756) X_174 = YFF CENTER OF CONGRUENCE (0.5941, 0.2335) X_176 = EQUAL DETOUR POINT (0.5929, 0.2323) X_3082 = RADICAL CENTER OF EXTERNAL MALFATTI CIRCLES (0.4418, 0.0738)
A
B
C
X8
X178
X174
X176
X3082
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) { 8, 178 }: Central Line L(8, 178) { 174, 176 }: Central Line L(174, 176)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) X_8 = NAGEL POINT (0.4024, 0.0200) X_178 = 2nd MID-ARC POINT (0.5558, 0.1107) X_174 = YFF CENTER OF CONGRUENCE (0.6235, 0.1737) X_176 = EQUAL DETOUR POINT (0.6225, 0.1730) X_3082 = RADICAL CENTER OF EXTERNAL MALFATTI CIRCLES (0.3949, 0.0156)
A
B
C
X8
X178
X174
X176
X3082
Kimberling (6, 9, 13) | AB = 1
X3082 (RADICAL CENTER OF EXTERNAL MALFATTI CIRCLES ) 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 \(f(A,B,C) = 1/(1 - \cos A/2)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(4.611475589035\)
▸ Incident Central Lines2 lines
