X2090
1st STEVANOVIC PERSPECTOR
The bisector A
X1 of angle A meets the
A -excircle in two points; let
A' be the point closer to
I , and define
B' and
C' cyclically. Then triangle
A'B'C' is perspective to the
medial triangle , and the perspector is
X2090 . (Milorad Stevanovic,
Hyacinthos #8088 , 10/02/03)
5, 6, 7
3, 4, 5
Kepler
Kimberling
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 174 }: Central Line L(2, 174) { 8, 188 }: Central Line L(8, 188) { 177, 178 }: Central Line L(177, 178) { 9, 362 }: Central Line L(9, 362) { 85, 555 }: Central Line L(85, 555)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) M_a = Midpoint M_a of BC (0.8061, 0.2999) M_b = Midpoint M_b of CA (0.3061, 0.2999) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5374, 0.2000) X_174 = YFF CENTER OF CONGRUENCE (0.5813, 0.2549) X_8 = NAGEL POINT (0.4694, 0.1333) X_188 = 2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE (0.5245, 0.1802) X_177 = 1st MID-ARC POINT (0.5862, 0.2654) X_178 = 2nd MID-ARC POINT (0.5439, 0.2098) X_9 = MITTENPUNKT (0.5110, 0.1615) X_362 = CONGRUENT CIRCUMCIRCLES ISOSCELIZER POINT (0.5299, 0.2081) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5610, 0.2421) X_555 = (X(1),X(158))-ANSWER TO QUESTION A (0.5982, 0.2990) X_2090 = 1st STEVANOVIC PERSPECTOR (0.5155, 0.1725)
A
B
C
Ma
Mb
Mc
X2
X174
X8
X188
X177
X178
X9
X362
X85
X555
X2090
(a, b, c) = (5, 6, 7) | AB = 1
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 174 }: Central Line L(2, 174) { 8, 188 }: Central Line L(8, 188) { 177, 178 }: Central Line L(177, 178) { 9, 362 }: Central Line L(9, 362) { 85, 555 }: Central Line L(85, 555)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6400, 0.4800) M_a = Midpoint M_a of BC (0.8200, 0.2400) M_b = Midpoint M_b of CA (0.3200, 0.2400) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5467, 0.1600) X_174 = YFF CENTER OF CONGRUENCE (0.6119, 0.2308) X_8 = NAGEL POINT (0.4400, 0.0800) X_188 = 2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE (0.5282, 0.1331) X_177 = 1st MID-ARC POINT (0.6174, 0.2453) X_178 = 2nd MID-ARC POINT (0.5559, 0.1735) X_9 = MITTENPUNKT (0.5091, 0.1091) X_362 = CONGRUENT CIRCUMCIRCLES ISOSCELIZER POINT (0.5319, 0.1802) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5801, 0.2201) X_555 = (X(1),X(158))-ANSWER TO QUESTION A (0.6298, 0.2920) X_2090 = 1st STEVANOVIC PERSPECTOR (0.5141, 0.1246)
A
B
C
Ma
Mb
Mc
X2
X174
X8
X188
X177
X178
X9
X362
X85
X555
X2090
(a, b, c) = (3, 4, 5) | AB = 1
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 174 }: Central Line L(2, 174) { 8, 188 }: Central Line L(8, 188) { 177, 178 }: Central Line L(177, 178) { 9, 362 }: Central Line L(9, 362) { 85, 555 }: Central Line L(85, 555)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6180, 0.4859) M_a = Midpoint M_a of BC (0.8090, 0.2429) M_b = Midpoint M_b of CA (0.3090, 0.2429) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5393, 0.1620) X_174 = YFF CENTER OF CONGRUENCE (0.5941, 0.2335) X_8 = NAGEL POINT (0.4499, 0.0817) X_188 = 2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE (0.5238, 0.1346) X_177 = 1st MID-ARC POINT (0.5989, 0.2484) X_178 = 2nd MID-ARC POINT (0.5471, 0.1756) X_9 = MITTENPUNKT (0.5077, 0.1103) X_362 = CONGRUENT CIRCUMCIRCLES ISOSCELIZER POINT (0.5270, 0.1823) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5676, 0.2231) X_555 = (X(1),X(158))-ANSWER TO QUESTION A (0.6094, 0.2961) X_2090 = 1st STEVANOVIC PERSPECTOR (0.5120, 0.1262)
A
B
C
Ma
Mb
Mc
X2
X174
X8
X188
X177
X178
X9
X362
X85
X555
X2090
Kepler (1, √φ, φ) | AB = 1
< A B C >: Reference Triangle △ABC < M_a M_b M_c >: Medial Triangle △M_a M_b M_c
[ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 2, 174 }: Central Line L(2, 174) { 8, 188 }: Central Line L(8, 188) { 177, 178 }: Central Line L(177, 178) { 9, 362 }: Central Line L(9, 362) { 85, 555 }: Central Line L(85, 555)
A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6331, 0.2801) M_a = Midpoint M_a of BC (0.8166, 0.1400) M_b = Midpoint M_b of CA (0.3166, 0.1400) M_c = Midpoint M_c of AB (0.5000, 0.0000) X_2 = CENTROID (0.5444, 0.0934) X_174 = YFF CENTER OF CONGRUENCE (0.6235, 0.1737) X_8 = NAGEL POINT (0.4024, 0.0200) X_188 = 2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE (0.5215, 0.0586) X_177 = 1st MID-ARC POINT (0.6266, 0.1906) X_178 = 2nd MID-ARC POINT (0.5558, 0.1107) X_9 = MITTENPUNKT (0.5022, 0.0343) X_362 = CONGRUENT CIRCUMCIRCLES ISOSCELIZER POINT (0.5180, 0.1471) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5911, 0.1796) X_555 = (X(1),X(158))-ANSWER TO QUESTION A (0.6316, 0.2388) X_2090 = 1st STEVANOVIC PERSPECTOR (0.5048, 0.0532)
A
B
C
Ma
Mb
Mc
X2
X174
X8
X188
X177
X178
X9
X362
X85
X555
X2090
Kimberling (6, 9, 13) | AB = 1
X2090 (1st STEVANOVIC PERSPECTOR ) 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 \(f(A,B,C) = (\sin B/2 + \sin C/2) \csc A\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(3.356486961348\)
▸ Incident Central Lines6 lines
{2 174 8 188 9 362 85 555 177 178 312 556
▸ Functional Relations2 relations
