There are two circles that are tangent to the circumcircle, NPC and incircle of a triangle ABC. The first one, ΓI, is internally tangent to NPC; the other, ΓE, is externally tangent to NPC.
NOTE: All coordinates are barycentrics.
1. Internal tangent circle ΓI:
Radius= `r*|(r^2-s^2+5*r*R+4*R^2)/(4*r*R+4*R^2+3*r^2-s^2)|`
Center Zi:
Zi = `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 Zi :
X = `(2*a^5-2*(b+c)*a^4+2*a^3*b*c+(b^2-c^2)*(b-c)*a^2-2*(b^2-c^2)^2*a+(b^2-c^2)^2*(b+c))/a/(b-c)/(-a+b+c) `: :
= 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 Zi:
X=`(b-c)^2*(a-b-c)*(a^2-b^2-c^2)*(a^10+(3*b*c-b^2-c^2)*a^8- 7*b*c*(b+c)*a^7+(b^3*c+22*b^2*c^2-2*c^4+b*c^3-2*b^4)*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^2-3*b*c+b^2)*(c^2+6*b*c+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 ΓE
Radius = `r*|(r^2-s^2+5*r*R+4*R^2)/(3*r^2-s^2+12*r*R)|`
Center ZE:
ZE = `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 ZE :
X = `2*a^9+12*a^7*b*c-4*(b+c)*a^8+(b+c)*(5*b^2-14*b*c+5*c^2)*a^6+(-5*b^4+12*b^2*c^2-5*c^4)*a^5+(b^2-c^2)*(b-c)*(2*b+c)*(b+2*c)*a^4+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*(3*b^2-2*b*c+3*c^2)*a^2-(b^2-c^2)^3*(b-c)*b*c+(b^4-c^4)*(b^2-c^2)*(b-c)^2*a` : :
= On lines: (1,5), (104,105), (106,676), (1320,2398)
= ( 7.254687161947005, 11.51554264197544, -7.679951421897236 )
Reflection of X(105) in ZE:
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*b^2+2*b*c+7*c^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*b^4-6*b^3*c+6*b^2*c^2-6*b*c^3+5*c^4)*a^2+(3*b^2+4*b*c+3*c^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 ΓI, ΓE
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, ΓAI is internally tangent to NPC, and the other, ΓAE, is externally tangent to NPC. Let OAI and OAE be their centers and TAIand TAE their points of contact with the circumcircle. Build and name other circles cyclically. Then:
· ΔOAIOBIOCIis perspective to EXCENTRAL and FEUERBACH triangles at X(5)
· ΔOAE OBEOCE 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 (TAITAE), (TBI TBE), (TCI TCE) concur at X(25)
· ΔTAITBITCIis perspective to CIRCUMPERP1 and FEUERBACH triangles at X(4220) and X(2), respectively.
· ΔTAETBETCEis 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
