Tangent circles - Conconic circumcenters

César E. Lozada
 

Let ABC be a triangle and P a point.

Denote:

Ba: the center of the circle through B and tangent to AP at P, and cyclically Cb, Ac

Ca: the center of the circle through C and tangent to AP at P, and cyclically Ab, Bc

 

Then six points Ba, Ca, Cb, Ab, Ac, Bc lie on a conic

 

-------------

 

For P=u:v:w (trilinears), the center Q(P) of the conic is;

Q(P) =  a*b^2*c*(6*a^4+2*b^4-8*a^2*b^2-b^2*c^2-5*a^2*c^2-c^4)*v^4*w*u^2+a*b*c^2*(-b^2*c^2+6*a^4-5*a^2*b^2+2*c^4-b^4-8*a^2*c^2)*v*w^4*u^2+a^2*b*c*(-3*a^2*b^2-3*a^2*c^2-2*b^2*c^2+b^4+c^4+2*a^4)*v*w*u^5-a^2*b*c^3*(-a^2+b^2+c^2)*v*w^5*u-4*a^2*b*c*(-a^2*c^2+a^4+3*b^2*c^2-a^2*b^2)*v^3*w^3*u-a^2*b^3*c*(-a^2+b^2+c^2)*v^5*w*u+2*a^3*w^2*v^5*b^3*c^2+2*a^3*w^5*v^2*b^2*c^3+(-2*a^2*b^4*c^2+12*a^6*b^2-12*a^4*b^4+12*b^2*c^6+8*a^2*c^6-3*b^8-2*a^2*b^2*c^4-5*a^8-12*a^4*c^4+8*a^2*b^6-18*b^4*c^4+12*b^6*c^2-3*c^8+12*a^6*c^2-2*a^4*b^2*c^2)*w^2*u^3*v^2-a^2*b^2*(a^4-2*a^2*b^2-a^2*c^2+b^4-b^2*c^2)*u^5*w^2-a^2*c^2*(a^4-a^2*b^2-2*a^2*c^2-b^2*c^2+c^4)*v^2*u^5-a^2*b^2*(a^4-2*a^2*b^2-b^2*c^2-9*a^2*c^2+b^4+8*c^4)*v^4*w^2*u-2*c*a*(-6*a^4*b^2-4*b^2*c^4-a^2*c^4+5*a^2*b^2*c^2-a^4*c^2+7*a^2*b^4+c^6+a^6-2*b^6+5*b^4*c^2)*v^2*w*u^4+a^2*b^2*c^2*(-a^2+b^2+c^2)*v^4*u^3-a^2*c^2*(c^4+8*b^4+a^4-b^2*c^2-2*a^2*c^2-9*a^2*b^2)*v^2*w^4*u-a*b^3*c^2*(a^2-b^2+c^2)*v^5*u^2-a*b^2*c^3*(a^2+b^2-c^2)*w^5*u^2-2*a*b*(-6*a^4*c^2-5*c^6+9*a^2*c^4-6*b^4*c^2+2*a^2*b^4+2*a^2*b^2*c^2-4*a^4*b^2+2*a^6+11*b^2*c^4)*w^2*u^2*v^3-2*c*a*(-6*b^2*c^4-5*b^6+11*b^4*c^2-6*a^4*b^2+9*a^2*b^4+2*a^6+2*a^2*b^2*c^2-4*a^4*c^2+2*a^2*c^4)*v^2*u^2*w^3-2*a*b*(-2*c^6-4*b^4*c^2-6*a^4*c^2+5*a^2*b^2*c^2+7*a^2*c^4-a^4*b^2+a^6+b^6+5*b^2*c^4-a^2*b^4)*w^2*u^4*v+b*c*(-4*a^2*b^2*c^2-17*a^4*b^2-9*a^4*c^2-c^6+b^6+7*a^6+9*a^2*b^4+3*b^2*c^4-3*b^4*c^2+3*a^2*c^4)*u^3*w^3*v+b*c*(-9*a^4*b^2-17*a^4*c^2+c^6-4*a^2*b^2*c^2-b^6+3*a^2*b^4+9*a^2*c^4+3*b^4*c^2+7*a^6-3*b^2*c^4)*u^3*w*v^3+a*b^2*c*(-3*a^2*b^2-3*a^2*c^2+2*c^4+a^4+2*b^4-4*b^2*c^2)*w^3*u^4+a*b*c^2*(-3*a^2*b^2-3*a^2*c^2+2*c^4+a^4+2*b^4-4*b^2*c^2)*v^3*u^4+a^2*b^2*c^2*(-a^2+b^2+c^2)*w^4*u^3+a^3*b*c^2*(-a^2+b^2+c^2)*v^3*w^4+a^3*b^2*c*(-a^2+b^2+c^2)*v^4*w^3 : :

 

ETC pairs (P, Q(P)): {2, 5569}, {4, 3818}, {13, 22846}, {14, 22891}, {74, 12041}, {98, 12042}, {99, 33813}, {100, 33814}, {110, 1511}, {111, 14650}, {3413, 3413}, {3414, 3414}

 

Some others:

 

Q( X(1) ) = MIDPOINT OF X(15600) AND X(15601)

= a*(5*a^2-4*(b+c)*a+3*(b-c)^2) : : (barys)

= 3*X(1)+X(3973), 3*X(1)+2*X(8692), 2*X(1)+X(15601), 2*X(3973)+3*X(15600), 2*X(3973)-3*X(15601), 4*X(8692)+3*X(15600), 4*X(8692)-3*X(15601)

= lies on these lines: {1, 6}, {55, 5573}, {57, 902}, {106, 1292}, {145, 17338}, {269, 1319}, {528, 4859}, {551, 4648}, {614, 2177}, {968, 29818}, {999, 21002}, {1149, 2293}, {1253, 7962}, {1388, 1456}, {1418, 13462}, {1420, 2263}, {1458, 33633}, {1471, 3340}, {1480, 18443}, {1621, 3677}, {1697, 28082}, {2646, 15839}, {2999, 3748}, {3052, 10980}, {3058, 23681}, {3158, 5272}, {3332, 13464}, {3333, 4257}, {3445, 30389}, {3601, 28011}, {3616, 17282}, {3620, 3883}, {3622, 3662}, {3636, 4349}, {3679, 17337}, {3744, 10582}, {3749, 5437}, {3755, 8236}, {3886, 17117}, {3915, 11518}, {3928, 8616}, {3938, 7308}, {3945, 17274}, {3957, 14997}, {4000, 30331}, {4421, 8056}, {4428, 4906}, {4512, 17597}, {4666, 5269}, {4684, 20080}, {4702, 17151}, {4779, 28557}, {4902, 28534}, {5284, 7322}, {5853, 16020}, {7982, 13329}, {9581, 28027}, {10385, 24177}, {11529, 21059}, {14996, 29817}, {17245, 25055}, {19624, 25415}, {24175, 34607}, {24841, 25728}, {26728, 31162}, {30117, 31393}

= midpoint of X(15600) and X(15601)

= reflection of X(i) in X(j) for these (i,j): (3973, 8692), (15600, 1)

= X(21)-Beth conjugate of-X(3243)

= intersection, other than A,B,C, of conics {{A, B, C, X(44), X(2191)}} and {{A, B, C, X(105), X(3243)}}

= X(10002)-of-2nd circumperp triangle

= X(15600)-of-5th mixtilinear triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 238, 3243), (1, 1001, 7174), (1, 1279, 7290), (1, 5223, 4864), (3749, 29820, 5437), (3973, 8692, 15601), (5272, 17715, 3158)

= [ 7.9233211380798410, 3.4862613007551300, -2.4298184823444850 ]

 

Q( X(3) ) = X(3)X(66) ∩ X(6)X(186)

= a^2*(a^10-(b^2+c^2)*a^8-2*(b^4+b^2*c^2+c^4)*a^6+(b^2+c^2)*(2*b^4+b^2*c^2+2*c^4)*a^4+(b^4-c^4)^2*a^2-(b^4-c^4)*(b^2-c^2)*(b^4+b^2*c^2+c^4)) : : (barys)

= (SB+SC)*((2*R^2-SW)*S^2-(12*R^2+SA-4*SW)*SA*SW) : :

= 3*X(3)+X(159), 4*X(3)-X(15579), 4*X(3)+X(15580), 5*X(3)+X(15581), 2*X(3)+X(15582), X(159)-3*X(15577), 2*X(159)+3*X(15578), 4*X(159)+3*X(15579), 4*X(159)-3*X(15580), 5*X(159)-3*X(15581), 2*X(159)-3*X(15582), 2*X(15577)+X(15578), 4*X(15577)+X(15579), 4*X(15577)-X(15580), 5*X(15577)-X(15581), 2*X(15578)+X(15580), 5*X(15578)+2*X(15581)

= lies on these lines: {2, 18382}, {3, 66}, {6, 186}, {20, 28408}, {22, 10192}, {24, 5480}, {26, 29181}, {69, 10298}, {140, 20300}, {154, 6636}, {161, 7485}, {182, 9977}, {206, 1511}, {297, 18380}, {376, 2916}, {378, 20987}, {511, 1658}, {524, 18324}, {549, 23300}, {578, 32191}, {631, 2917}, {1092, 32391}, {1350, 7488}, {1853, 15246}, {2070, 31267}, {2393, 5092}, {2854, 12893}, {2883, 10323}, {3564, 15331}, {3589, 6644}, {3763, 34775}, {3818, 18570}, {3827, 13624}, {5085, 22467}, {5447, 7525}, {5893, 11414}, {5895, 16661}, {6000, 33533}, {6697, 18400}, {6759, 15067}, {6776, 21844}, {7492, 10117}, {7512, 17821}, {7514, 34573}, {7550, 18405}, {7575, 21850}, {7998, 15139}, {8546, 18571}, {8550, 32534}, {9682, 13910}, {9821, 15257}, {9924, 10249}, {9969, 11430}, {10182, 12106}, {10201, 23306}, {11178, 32600}, {11250, 29012}, {12007, 15750}, {12017, 34777}, {13347, 32184}, {13367, 19161}, {14649, 18472}, {15040, 15141}, {15462, 18438}, {16063, 23315}, {17714, 29317}, {17847, 33884}, {17907, 18121}, {18916, 19468}, {19153, 33878}, {21213, 23292}, {22109, 33851}, {29323, 32903}, {32274, 32607}, {32598, 32609}

= midpoint of X(i) and X(j) for these {i,j}: {3, 15577}, {206, 3098}, {1350, 34117}, {10282, 14810}, {15578, 15582}, {15579, 15580}

= reflection of X(i) in X(j) for these (i,j): (15578, 3), (15579, 15578), (15580, 15582), (15582, 15577), (20300, 140)

= complement of X(18382)

= X(15578)-of-ABC-X3 reflections triangle

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 15582, 15579), (161, 7485, 23332), (1350, 23041, 34117), (3098, 11202, 206), (15577, 15578, 15580), (17821, 31884, 19149)

= [ 4.4004245476072550, 3.0305201728859910, -0.4883531212938852 ]

 

Q( X(15) ) = X(3)X(6) ∩ X(18)X(531)

= a^2*(2*sqrt(3)*(3*a^2-2*b^2-2*c^2)*S+a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4) : : (barys)

= 3*X(15)+X(5237), X(5237)-3*X(30560)

= lies on these lines: {3, 6}, {18, 531}, {6109, 16964}, {6781, 16001}, {7749, 16002}, {8594, 33274}, {22510, 31709}

= midpoint of X(15) and X(30560)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 22236, 3104), (50, 13333, 15797), (500, 15796, 4280), (572, 13331, 22052), (1380, 11916, 9690), (1805, 8410, 9786), (2673, 4263, 9994), (3312, 32447, 11482), (3368, 15883, 1687), (3386, 5024, 1684), (3386, 32110, 4286), (3394, 22425, 10542), (4266, 8410, 3581), (4271, 18114, 15905), (6441, 15167, 18994), (22052, 35006, 3284)

= [ -28.4490739694097300, -20.9150368986556800, 31.2506472437812700 ]

 

Q( X(16) ) = X(3)X(6) ∩ X(17)X(530)

= a^2*(-2*sqrt(3)*(3*a^2-2*b^2-2*c^2)*S+a^4-3*(b^2+c^2)*a^2+2*b^4-2*b^2*c^2+2*c^4) : : (barys)

= 3*X(16)+X(5238), X(5238)-3*X(30559)

= lies on these lines: {3, 6}, {17, 530}, {6108, 16965}, {6781, 16002}, {7749, 16001}, {8595, 33274}, {22511, 31710}

= midpoint of X(16) and X(30559)

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 22238, 3105), (2019, 33871, 12055), (15012, 15851, 22811)

= [ -4.1327458045678710, -2.4110949623289710, 7.2173052117819150 ]

 

Q( X(1113) )  = MIDPOINT OF X(3) AND X(1113)

= a*((4*a^4-3*(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^6-(b^2+c^2)*a^4-(b^4-3*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*b*c-2*(2*a^8-4*(b^2+c^2)*a^6+12*a^4*b^2*c^2+(b^2+c^2)*(4*b^4-11*b^2*c^2+4*c^4)*a^2-(b^2-c^2)^2*(b^2+2*c^2)*(2*b^2+c^2))*S*OH*a) : : (barys)

= 2*OH*R*(3*S^2-5*SB*SC)+a^2*(S^2-(42*R^2-SA-8*SW)*SA) : : (barys)

= 3*X(2)+X(15160), 3*X(3)-X(1114), 3*X(3)+X(15154), 5*X(3)-X(15155), 7*X(3)-X(15156), 5*X(3)+X(15157), X(3)+3*X(28447), 7*X(3)-3*X(28448), 2*X(3)+X(30524), 4*X(3)-X(30525), 3*X(376)+X(14808), 3*X(381)-X(10736), 2*X(548)+X(20408), 5*X(631)-X(14807), 3*X(1113)+X(1114), 3*X(1113)-X(15154), 5*X(1113)+X(15155), 7*X(1113)+X(15156), 5*X(1113)-X(15157), X(1113)-3*X(28447), 7*X(1113)+3*X(28448), 4*X(1113)+X(30525)

= lies on these lines: {2, 3}, {1511, 2574}, {2100, 3576}, {2102, 10246}, {2103, 12702}, {2104, 5050}, {2105, 33878}, {2575, 12041}, {5085, 15162}, {7740, 10288}, {14500, 16111}

= midpoint of X(i) and X(j) for these {i,j}: {3, 1113}, {20, 10751}, {1114, 15154}, {1657, 10737}, {2103, 12702}, {2105, 33878}, {3534, 10720}, {10750, 15160}, {14500, 16111}, {15155, 15157}

= reflection of X(i) in X(j) for these (i,j): (1312, 31681), (1313, 140), (13627, 549), (20409, 31682), (30524, 1113), (31682, 3530)

= complement of X(10750)

= circumperp conjugate of X(15155)

= X(523)-vertex conjugate of-X(15154)

= circumcircle-inverse of-X(15154)

= X(10736)-of-Ehrmann-mid triangle

= [ 3.5867191079950600, 2.7077373295270680, 0.1106682808525353 ]

 

Q( X(1114) )  = MIDPOINT OF X(3) AND X(1114)

= a*((4*a^4-3*(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^6-(b^2+c^2)*a^4-(b^4-3*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*b*c+2*(2*a^8-4*(b^2+c^2)*a^6+12*a^4*b^2*c^2+(b^2+c^2)*(4*b^4-11*b^2*c^2+4*c^4)*a^2-(b^2-c^2)^2*(b^2+2*c^2)*(2*b^2+c^2))*S*OH*a) : : (barys)

= -2*OH*R*(3*S^2-5*SB*SC)+a^2*(S^2-(42*R^2-SA-8*SW)*SA) : : (barys)

= 3*X(2)+X(15161), 3*X(3)-X(1113), 5*X(3)-X(15154), 3*X(3)+X(15155), 5*X(3)+X(15156), 7*X(3)-X(15157), 7*X(3)-3*X(28447), X(3)+3*X(28448), 4*X(3)-X(30524), 2*X(3)+X(30525), 3*X(376)+X(14807), 3*X(381)-X(10737), 2*X(548)+X(20409), 5*X(631)-X(14808)

= lies on these lines: {2, 3}, {1511, 2575}, {2101, 3576}, {2102, 12702}, {2103, 10246}, {2104, 33878}, {2105, 5050}, {2574, 12041}, {5085, 15163}, {7740, 10287}, {14499, 16111}

= midpoint of X(i) and X(j) for these {i,j}: {3, 1114}, {20, 10750}, {1113, 15155}, {1657, 10736}, {2102, 12702}, {2104, 33878}, {3534, 10719}, {10751, 15161}, {14499, 16111}, {15154, 15156}

= reflection of X(i) in X(j) for these (i,j): (1312, 140), (1313, 31682), (13626, 549), (20408, 31681), (30525, 1114), (31681, 3530)

= complement of X(10751)

= circumperp conjugate of X(15154)

= X(523)-vertex conjugate of-X(15155)

= circumcircle-inverse of-X(15155)

= [ 9.9780066803976270, 9.0821645239357340, -7.2522217333084980 ]

 

César Lozada

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