Let ABC be a triangle, Am, Bm, Cm the midpoints of their sides, (O) its circumcircle and A* a point on the tangent Ta to (O) through A.
Denote:
(Ab): the circle through C tangent to A*Bm at Bm
(Ac): the circle through B tangent to A*Cm at Cm
Then one of the intersections of (Ab), (Ac) lies on BC. (Let’s name it A’)
Jean Louis Ayme, http://jl.ayme.pagesperso-orange.fr/Docs/Intersection%20sur%20un%20cote.pdf, Dec. 31, 2018
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Application I:
Let P be a point in the plane of ABC and A* the point where the parallel to BC through P cuts Ta..
Define B*, C*, Tb, Tc, (Bc), (Ba), (Ca), (Cb), B’, C’ cyclically.
Then:
1) The locus of P such that AA’, BB’, CC’ are concurrent is the line at infinity or the hyperbola with center and perspector X(11794) and through ETC centers X(39) and X(216). For these centers the points of concurrence Q1 of the given lines are:
Q1( X(39) ) = X(2)X(3613) ∩ X(98)X(251)
= (a^2+b^2)*(a^2+c^2)*((b^2+c^2)*a^2+b^2*c^2-c^4)*((b^2+c^2)*a^2-b^4+b^2*c^2) : : (barys)
= on Kiepert hyperbola and lines: {2, 3613}, {76, 3060}, {83, 14957}, {98, 251}, {1916, 11794}, {2052, 10550}
= isotomic conjugate of the anticomplement of X(20965)
= barycentric product X(i)*X(j) for these {i,j}: {83, 3613}, {308, 27375}
= barycentric quotient X(i)/X(j) for these (i,j): (32, 3203), (82, 18042), (83, 1078), (251, 5012)
= trilinear product X(82)*X(3613)
= trilinear quotient X(i)/X(j) for these (i,j): (31, 3203), (82, 5012), (83, 18042)
= [ -1..5266201024090860, -1.6463305673936550, 5.4850256912149480 ]
Q1( X(216) ) = EULER LINE INTERCEPT OF X(51)X(324)
= (a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^4-b^2*c^2-(b^2+c^2)*a^2)*((b^2+c^2)*a^2-b^4+2*b^2*c^2-c^4) : : (barys)
= on lines: {2, 3}, {51, 324}, {53, 17500}, {110, 275}, {143, 14978}, {251, 6531}, {264, 3060}, {1629, 5012}, {2052, 5640}, {3289, 6748}, {6747, 19130}, {6749, 25051}, {8884, 13434}, {11451, 15466}, {14389, 19174}
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 458, 14957), (4, 6819, 1370), (5, 6755, 467)
= barycentric product X(i)*X(j) for these {i,j}: {53, 1078}, {308, 27370}, {311, 10312}, {324, 5012}, {343, 1629}
= barycentric quotient X(i)/X(j) for these (i,j): (53, 3613), (1629, 275)
= trilinear product X(i)*X(j) for these {i,j}: {53, 18042}, {1078, 2181}
= trilinear quotient X(i)/X(j) for these (i,j): (1629, 2190), (2181, 27375)
= [ 0.4980179340005647, -0.3728157727817433, 3.6689132781407090 ]
2) The locus of P such that the six circles (Ab), (Ac), (Bc), (Ba), (Ca), (Cb) are concurrent is the line at infinity or the rectangular hyperbola with center an perspector X(930) through ETC’s 5, 15704. For these centers the circles concur at Q2:
Q2( X(5) ) = X(5)
Q3( X(15704) ) = X(3)X(21357) ∩ X(5)X(10721)
= (4*SA-143*R^2+30*SW)*S^2+11*(11*R^2-2*SW)*SB*SC : : (barys)
= X(15704)+2*X(17505)
= on lines: {3, 21357}, {5, 10721}, {368, 8421}, {549, 12162}, {550, 20191}, {15704, 17505}
= [ 6.7343486634225100, 6.0827103407906160, -3.6786035986810520 ]
Application II:
Let P be a point in the plane of ABC and A* the orthogonal projection of P on Ta.
Define B*, C*, Tb, Tc, (Bc), (Ba), (Ca), (Cb), B’, C’ cyclically.
Then:
1) The locus of P such that AA’, BB’, CC’ are concurrent is the line at infinity or a conic with center O1=X(3) and through ETC centers X(1379) and X(1380). For these centers the points of concurrence Q1 of the given lines are:
Q1( X(1379) ) = ANTICOMPLEMENT OF X(13636)
= (-(2*a^2-b^2-c^2)*K+2*a^4-2*(b^2+c^2)*a^2-b^4+4*b^2*c^2-c^4)*(b^2-c^2-K)*(-a^2+b^2-K)*(c^2-a^2-K)*(-b^2+c^2-K) : :, where K=sqrt(SW^2-3*S^2) (barys)
= on Kiepert parabola, cubics K010 (Simson cubic), K015, K242, K408 and these lines: {2, 1340}, {99, 110}, {523, 6190}
= anticomplement of X(13636)
= isotomic conjugate of the anticomplement of X(13722)
= trilinear pole of the line {115, 2029}
= reflection of X(2) in the line X(3413)X(9168)
= barycentric product X(i)*X(j) for these {i,j}: {670, 2029}, {2966, 14501}, {3414, 6190}
= barycentric quotient X(i)/X(j) for these (i,j): (512, 2028), (1379, 1380), (2029, 512), (2799, 14502), (3414, 3413)
= trilinear product X(799)*X(2029)
= trilinear quotient X(i)/X(j) for these (i,j): (661, 2028), (2029, 798)
= [ 0.5418637313200825, -0.8117268511145955, 3.9525382720698300 ]
Q1( X(1380) ) = ANTICOMPLEMENT OF X(13722)
= ((2*a^2-b^2-c^2)*K+2*a^4-2*(b^2+c^2)*a^2-b^4+4*b^2*c^2-c^4)*(b^2-c^2+K)*(-a^2+b^2+K)*(c^2-a^2+K)*(-b^2+c^2+K) : :, where K=sqrt(SW^2-3*S^2) (barys)
= on Kiepert parabola, cubics K010 (Simson cubic), K015, K242, K408 and these lines: {2, 1341}, {99, 110}, {523, 6189}
= anticomplement of X(13722)
= isotomic conjugate of the anticomplement of X(13636)
= trilinear pole of the line {115, 2028}
= reflection of X(2) in the line X(3414)X(9168)
= barycentric product X(i)*X(j) for these {i,j}: {670, 2028}, {2966, 14502}, {3413, 6189}
= barycentric quotient X(i)/X(j) for these (i,j): (512, 2029), (1380, 1379), (2028, 512), (2799, 14501), (3413, 3414)
= trilinear product X(799)*X(2028)
= trilinear quotient X(i)/X(j) for these (i,j): (661, 2029), (2028, 798)
= [ 12.7588909301436000, -3.7853895405115830, 0.3725998883492020 ]
2) The locus of P such that the six circles (Ab), (Ac), (Bc), (Ba), (Ca), (Cb) are concurrent is the line at infinity or the conic with center X(12038) through ETC’s 110, 12901, 23181. For these centers the circles concur at:
Q2( X(110) ) = X(110)X(351) ∩ X(476)X(1304)
= (SB+SC)*(SA-SB)*(SA-SC)*(SA-15*R^2+3*SW) : : (barys)
= on lines: {3, 6030}, {23, 16186}, {30, 14385}, {110, 351}, {476, 1304}, {1113, 10288}, {1114, 10287}, {2070, 14670}, {2071, 7740}, {6760, 12113}, {9717, 15107}, {10130, 11058}, {12270, 14703}, {13595, 18114}, {14685, 15080}
= [ 0.6218724484277477, -0.7132977838090584, 3.8474679714239980 ]
Q2( X(12901) ) = X(476)X(2407) ∩ X(9033)X(12901)
= (SB-SC)*(4*S^2-3*R^2*(30*R^2+5*SA-17*SW)+3*SA^2-4*SB*SC-7*SW^2) : : (barys)
= on lines: {67, 9003}, {476, 2407}, {523, 550}, {924, 12162}, {2528, 3313}, {9033, 12901}
= [ 69.5891517320162100, -44.8979050524735800, 2.6057594880739990 ]
Q2( X(23181) ) = EULER LINE INTERCEPT OF X(476)X(2407)
= (SA-SB)*(SA-SC)*(2*S^2-(SB+SC)*(SA-6*R^2+2*SW)) : : (barys)
= on lines: {2, 3}, {107, 13398}, {110, 925}, {476, 2407}, {691, 16167}, {1302, 3565}, {3233, 5502}, {5468, 6563}
= [ 0.5434476775901084, -0..3275058740028750, 3.6165773127140120 ]
Application III:
Let P be a point in the plane of ABC and A* the point where Ta is cut by the polar trilinear of P.
Define B*, C*, Tb, Tc, (Bc), (Ba), (Ca), (Cb), B’, C’ cyclically.
Then
1) The locus of P such that AA’, BB’, CC’ are concurrent is a circum-sixtic through the vertices of the Thomson triangles and these ETC centers: {2, 4, 1000, 3431}. For these centers the points of concurrence Q1 of the given lines are:
Q1( X(2) ) = X(2)
Q1( X(4) ) = X(4)
Q1( X(3431) ) = X(4846)
Q1( X(1000) ) = ISOGONAL CONJUGATE OF X(1470)
= (-a+b+c)*(a^3-(b-c)*a^2-(b^2-2*b*c-c^2)*a+(b^2-c^2)*(b-c))*(a^3+(b-c)*a^2+(b^2+2*b*c-c^2)*a+(b^2-c^2)*(b-c)) : : (barys)
= on the Feuerbach hyperbola and thes lines: {1, 908}, {2, 104}, {4, 5554}, {7, 5080}, {9, 6735}, {10, 90}, {21, 2551}, {65, 5555}, {80, 3434}, {84, 377}, {149, 24297}, {388, 1476}, {390, 13278}, {404, 12667}, {405, 10942}, {442, 18542}, {443, 7705}, {452, 943}, {497, 1320}, {958, 10958}, {1000, 3421}, {1001, 10956}, {1041, 1877}, {1156, 2550}, {1329, 22768}, {1389, 5046}, {1392, 4345}, {1478, 3306}, {1512, 3359}, {1519, 6957}, {1537, 6929}, {1837, 10522}, {2320, 5328}, {2346, 11239}, {2475, 10308}, {2481, 11185}, {3577, 26333}, {3680, 5727}, {3753, 18516}, {3897, 5084}, {4187, 16203}, {4190, 18491}, {5187, 10532}, {5250, 7162}, {5251, 6910}, {5559, 5692}, {6850, 25005}, {6872, 11248}, {6919, 10586}, {6930, 12775}, {6931, 7951}, {8068, 10584}, {10527, 26476}, {10596, 14497}, {10679, 11113}, {11108, 18545}, {12608, 19860}, {12647, 18254}, {20895, 30479}, {21301, 23836}
= isogonal conjugate of X(1470)
= trilinear pole of the line {650, 2804}
= {X(6256), X(24982)}-harmonic conjugate of X(377)
= barycentric product X(312)*X(998)
= barycentric quotient X(i)/X(j) for these (i,j): (8, 17740), (9, 997), (21, 26637), (607, 11383), (650, 9001), (998, 57)
= trilinear product X(i)*X(j) for these {i,j}: {8, 998}, {522, 9058}
= trilinear quotient X(i)/X(j) for these (i,j): (8, 997), (29, 4227), (33, 11383), (312, 17740), (333, 26637), (522, 9001), (998, 56)
= [ 3.0274897416887420, 2.3034418294366570, 0.6486710269026590 ]
2) The locus of P such that the six circles (Ab), (Ac), (Bc), (Ba), (Ca), (Cb) are concurrent is a circum-sixtic through ETC’s {2, 6, 2052}. For these centers the circles concur at Q2:
Q2( X(2) ) = X(3)
Q2( X(6) ) = X(2)
Q2( X(2052) ) = X(4)
César Lozada
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