Tangent circles to circumcircle, NPC and incircle (with extraversions).

There are two circles that are tangent to the circumcircle, NPC and incircle of a triangle ABC. The first one, Γ_I, is internally tangent to NPC; the other, Γ_E, is externally tangent to NPC.

NOTE: All coordinates are barycentrics.

1.    Internal tangent circle Γ_I:

Radius= r*|(r^2-s^2+5*r*R+4*R^2)/(4*r*R+4*R^2+3*r^2-s^2)|

Center Z_i:

Z_i = 2*a^6-2*(b+c)*a^5-(b^2-4*b*c+c^2)*a^4+2*(b^2-c^2)*(b-c)*a^3+4*(b^2-c^2)*(b-c)*a*b*c-2*(b^2+3*b*c+c^2)*(b-c)^2*a^2+(b^2-c^2)^2*(b-c)^2 : :

= Complement of X(2968)

= Midpoint of: (1897,2968)                                                                        

= On lines:  (1,5), (2,1897), (3,108), (30,1785), (105,5020), (165,1360), (521,3042), (676,2804), (867,1862), (1068,3149), (1532,1870), (1936,5762), (2635,6357)

= (4*R^2-SW)*X(1)+3*r^2*X(2)-r^2*X(3)

= ( 1.611730809126064, 1.55156088792320, 1.822631186056290 )

Passes through X(11) y X(108), having antipodes:

    Reflection of X(11) in Z_i :

    X = 2*a^6-2*(b+c)*a^5-(b^2-4*b*c+c^2)*a^4+2*(b^2-c^2)*(b-c)*a^3+4*(b^2-c^2)*(b-c)*a*b*c-2*(b^2+3*b*c+c^2)*(b-c)^2*a^2+(b^2-c^2)^2*(b-c)^2 : :

       = Midpoint of: (100,1897)                                                                         

       = Reflection of: (2968/3035)                                                                        

      = On lines: (1,5), (100,108), (109,900), (523,2222), (2149,3700), (2968,3035)

      = ( 0.580326497006925, -0.26962876949254, 3.559487554476018 )

     Reflection of X(108) in Z_i:

X=(b-c)^2*(-b-c+a)*(-b^2-c^2+a^2)*(a^10+(3*b*c-c^2-b^2)*a^8-7*b*c*(b+c)*a^7+(-2*c^4+22*b^2*c^2-2*b^4+b*c^3+b^3*c)*a^6+b*c*(b+c)*(9*c^2-22*b*c+9*b^2)*a^5+(2*b^4-5*b^3*c-18*b^2*c^2-5*b*c^3+2*c^4)*(b-c)^2*a^4+3*(b^2-c^2)*(b-c)*b*c*a^3*(c^2+6*b*c+b^2)+(b^2-c^2)^2*a^2*(c^2+6*b*c+b^2)*(-3*b*c+c^2+b^2)-5*(b^2-c^2)^3*(b-c)*a*b*c-(b^2-c^2)^4*(b-c)^2) : :

= Reflection of: (2968/123)                                                                         

= On lines:  (3,108), (11,123), (1364,2850)

= ( 3.305222351136989, 2.97411313557093, 0.056175841372050 )

2.    External tangent circle Γ_E

Radius = r*|(r^2-s^2+5*r*R+4*R^2)/(3*r^2-s^2+12*r*R)|

Center Z_E:

Z_E = 2*a^6-2*(b+c)*a^5-(b^2-4*b*c+c^2)*a^4+2*(b^2-c^2)*(b-c)*a^3-2*(b^2-b*c+c^2)*(b-c)^2*a^2-4*(b^2-c^2)*(b-c)*a*b*c+(b^2-c^2)^2*(b-c)^2 : :

= On lines:  (1,5), (3,105), (108,1598), (238,5762), (516,3246), (517,3008), (676,2826),  (948,999),                             (1360,3361), (4310,5779)

= SW*X(1)-3*(4*R*r+r^2)*X(2)+(4*R*r+r^2)*X(3)

= ( 4.948911141596107, 7.44414659365719, -3.797088302130342 )

Passes through X(11) y X(105), having antipodes:

    Reflection of X(11) in Z_E :

       X =2*a^9-4*(b+c)*a^8+12*a^7*b*c+(b+c)*(5*c^2-14*b*c+5*b^2)*a^6+(-5*c^4+12*b^2*c^2-5*b^4)*a^5+(b^2-c^2)*(b-c)*a^4*(b+2*c)*(2*b+c)+2*(b^4-3*b^3*c+b^2*c^2-3*b*c^3+c^4)*(b-c)^2*a^3-(b^2-c^2)*(b-c)^3*a^2*(3*c^2-2*b*c+3*b^2)+(b^4-c^4)*(b^2-c^2)*a*(b-c)^2-(b^2-c^2)^3*(b-c)*b*c : :

        = On lines: (1,5), (104,105), (106,676), (1320,2398)

        = ( 7.254687161947005, 11.51554264197544, -7.679951421897236 )

    Reflection of X(105) in Z_E:

X =(b-c)^2*(a^7-3*(b+c)*a^6+(5*b^2+9*b*c+5*c^2)*a^5-(b+c)*(7*c^2+2*b*c+7*b^2)*a^4+(7*b^4+4*b^3*c+4*b^2*c^2+4*b*c^3+7*c^4)*a^3-(b+c)*(5*c^4-6*b*c^3+6*b^2*c^2-6*b^3*c+5*b^4)*a^2+(3*c^2+4*b*c+3*b^2)*(b^2-b*c+c^2)*(b-c)^2*a-(b^4-c^4)*(b-c)^3) : :

= On lines:  (3,105), (11,1111), (1086,2820)

= ( 7.203893096245062, 5.53877424438016, -3.518745270161146 )

3.    Similitude centers of circles Γ_I, Γ_E

Insimil-center = X(11)

Exsimil-center

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

= On lines: (1,5), (25,105), (109,1086), (226,1386), (1331,5856), (1458,6357), (1465,3011), (2361,5762)

= (2*R^2-SW)*X(1)+2*(2*R*r+r^2)*X(5)

= ( -1.089194017590830, -3.21756437369972, 6.370913979510722 )  

4.    Extraversions

Taking A-excircle instead of incircle, we find also two tangent circles to A-excircle, circumcircle and NPC. One of them, Γ_AI is internally tangent to NPC, and the other,  Γ_AE, is externally tangent to NPC. Let O_AI and O_AE be their centers and T_AI and T_AE their points of contact with the circumcircle. Build and name other circles cyclically.  Then:

·         ΔO_AIO_BIO_CI is perspective to EXCENTRAL and FEUERBACH triangles at X(5)

·         ΔO_AE O_BE O_CE is perspective to EXCENTRAL and FEUERBACH triangles at X(5) and it is also perspective to MEDIAL triangle at:

Z = (2*a^6+2*(b+c)*a^5-(b^2-4*b*c+c^2)*a^4-2*(b^2-c^2)*(b-c)*a^3-4*(b^2-c^2)*

(b-c)*a*b*c-2*(b^2+3*b*c+c^2)*(b-c)^2*a^2+(b^2-c^2)^2*(b-c)^2)*(a-b-c)^2

= Complement of X(6356)

 = On lines: (2,1119), (3,281), (5,9), (268,3560), (284,1146), (610,5787), (2322,2968), (3739,5745)

 = ( 4.713912920783893, 3.19721412780796, -0.748443570013695 )

·         Lines (T_AI T_AE), (T_BI T_BE), (T_CI T_CE) concur at X(25)

·         ΔT_AIT_BIT_CI is perspective to CIRCUMPERP1 and FEUERBACH triangles at X(4220) and X(2), respectively.

·         ΔT_AET_BET_CE is perspective to CIRCUMPERP2 and MIDARC triangles at X(28), to FEUERBACH triangle at X(4), and to TANGENTIAL triangle at:

Z= (-4*cos((B-C)/2)*sin(A/2)*cos(A)-cos(B-C)+2*cos(A)^2+cos(A))*sin(A)^2*sec(A)::

 = On lines: (1,19), (3,281), (4,198), (25,1604), (34,2270), (56,1249), (57,2331), (92,1817), (104,1436), (108,393), (579,1783), (910,1841), (1033,1617), (1400,2202), (1753,2324), (1826,4219), (1870,2262), (1880,1951), (2322,2975), (4224,5089)

 = ( -68.607417041979430, -92.43074522778095, 99.296142120515550 )

5.    Index of centers.

6-9-13 first coordinate	Point
-68.60741704197940	Perspector ΔTAETBETCE - TANGENTIAL triangle of ABC
-1.08919401759083	Exsimicenter of ΓI,  ΓE
0.58032649700693	Reflection of X(11) in ZI
1.61173080912606	Center of ΓI = ZI
3.30522235113698	Reflection of X(108) in ZI
4.71391292078389	Perspector ΔOAEOBEOCE - MEDIAL triangle of ABC
4.94891114159610	Center of ΓE = ZE
7.20389309624506	Reflection of X(105) in ZE
7.25468716194700	Reflection of X(11) in ZE

César Lozada

Feb 16, 2015

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