Thursday, February 19, 2015

FEUERBACH-TANGENTS AND APOLLONIUS TRIANGLES

FEUERBACH-TANGENTS AND APOLLONIUS TRIANGLES

Tangents circles of excircles of ΔABC are the nine-point-circle {N} and the Apollonius circle {Ap}.

Let ΔPAPBPC be the triangle having vertices at the contact points of {Ap} and the excircles and ΔTATBTC the triangle limited by the tangents to {Ap} at those points.

Similarly, let ΔT’AT’BT’C the triangle limited by the tangents to {N} through the points where it touches the excircles, i.e., the Feuerbach points of  ΔABC.

We could name ΔPAPBPC as APOLLONIUS-CONTACT-TRIANGLE of ΔABC, ΔTATBTC as APOLLONIUS-TANGENTS-TRIANGLE of ΔABC and ΔT’AT’BT’C as FEUERBACH-TANGENTS-TRIANGLE of ΔABC.

Next tables show relations between these and other named triangles.

Note: All coordinates are trilinear.


·         PERSPECTIVE TRIANGLES

Triangles T1 & T2
Perspector T1 & T2
APOLLONIUS-CONTACT
  ABC
X(181)
APOLLONIUS-CONTACT
  EXCENTRAL
X(970)
APOLLONIUS-CONTACT
  EXTANGENTS
a*(b+c)*(SA*(SA+SW)-(a+c)*SB*b-(a+b)*SC*c)
On lines: (1,3688), (10,12), (40,43), (55,386), (71,213), (218,573), (511,5247), (672,4300),  (674,1104), (958,4259), (960,4026), (1284,3191), (1362,4306), (1402,3682), (1453,3056), (1468,3917), (1724,3271), (2093,6048), (2292,3690), (2294,3954), (2841,3030), (2911,3556), (3032,6154), (3611,5360), (3869,4972)
-2.763868750807431
APOLLONIUS-CONTACT
  FEUERBACH
X(10)
APOLLONIUS-CONTACT
  INCENTRAL
X(1682)
APOLLONIUS-CONTACT
  SYMMEDIAL
X(2092)
APOLLONIUS-TANGENTS
  ABC
X(2092)
APOLLONIUS-TANGENTS
  MEDIAL
X(10)
APOLLONIUS-TANGENTS
  SYMMEDIAL
(b+c)*(SB+SC)*(SA-2*s^2-b*c)*(SA+2*s^2)
On lines: (6,10), (42,2300), (71,213), (181,1409), (1185,2273), (2347,4274), (3169,3293)
0.816253492362215
FEUERBACH-TANGENTS
  APOLLONIUS-TANGENTS
X(10)
FEUERBACH-TANGENTS
  CIRCUMPERP1
X(30)
FEUERBACH-TANGENTS
  FEUERBACH
X(5949)
FEUERBACH-TANGENTS
  LEMOINE
(b+c)/(a*(2*b^2+2*c^2-a^2))
On lines: (10,598), (313,4062), (1089,4527), (3993,4013)
 
1.113329655573671
FEUERBACH-TANGENTS
  MEDIAL
X(10)
FEUERBACH-TANGENTS
  ORTHIC
X(1826)



·         ORTHOLOGIC TRIANGLES

Triangles T1 & T2
T1-orthologic center of T2
T2-orthologic center of T1
APOLLONIUS-TANGENTS
 CIRCUMPERP1
X(10)
a*((2*a-s)*S^2-SA*(SW*s-3*a*b*c-S^2/s))
On lines: (1,1357), (3,595), (165,970), (511,3579), (517,548), (1158,2808), (3667,4075), (3871,3937)
1.073056961879668
APOLLONIUS-TANGENTS
 CIRCUMPERP2

APOLLONIUS-TANGENTS
 MIDARC
X(10)
a*(2*SA*(2*S*(R-r)+SW*s)-(3*a-b-c)*S^2)
Midpoint of: (1,970)
On lines: (1,181), (3,595), (40,1054), (51,3897), (140,517), (511,1385), (953,6083)
12.491668826513020
APOLLONIUS-TANGENTS
 EXCENTRAL
X(10)
X(970)
APOLLONIUS-TANGENTS
 HEXYL
X(10)
a*((2*R*S-s*SW)*SA+s*S^2)
Reflection of: (5/5482), (970/3)
On lines: (1,1401), (3,6), (5,5482), (20,5208), (21,3917), (51,404), (84,295), (405,3819), (474,5943), (517,550), (978,3271), (988,3056), (1012,5907), (1408,5347),           
 (2810,3811), (2979,4189), (3060,4188), (3794,4201), (3868,3937), (4349,5045), (5047,5650)
 -9.728303355176322
APOLLONIUS-TANGENTS
 HUTSON INTOUCH
X(10)
a*((7*a-b-c)*S^2+2*(3*a-s)*SA^2-2*(s-b)*SA*SB-2*(s-c)*SA*SC)
On lines: (1,1401), (51,3885), (970,1697), (2810,3244), (2841,3881)
0.389042200097107
APOLLONIUS-TANGENTS
 INTOUCH
X(10)
a*(SA^2*(s+a)+((s-b)*SB+(s-c)*SC)*SA+s*S^2)
Midpoint of: (1071,5907)
On lines:  (1,1401), (3,951), (10,2810), (21,3937), (40,4334), (57,970), (72,3819), (182,3157),
 (511,942), (517,4298), (1357,1682), (1385,2818), (2841,3884), (3868,3917), (3876,5650), (5439,5943), (5482,5719), (5708,5752)
2.991574818816960
FEUERBACH-TANGENTS
 CIRCUMPERP1
X(10)
(5*S^2-7*SB*SC-2*(a-c)*SB*b-2*(a-b)*SC*c)/a
Midpoint of: (20,5690), (40,550), (1385,5493)
Reflection of: (946/3530), (5901/3)
On lines: (3,962), (5,165), (8,3534), (10,30), (11,5442), (20,4678), (35,3649), (40,550), (79,4995), (100,3648), (140,516), (145,376), (477,901), (484,5441), (515,4746),    
 (517,548), (549,3624), (632,1699), (946,3530), (1293,2372), (1385,5493), (1482,3522), (1657,5657), (1698,3845), (2771,4127), (3474,6147), (3529,5790),    
 (3634,5066), (3650,4420), (4297,5844), (5073,5818)
12.443135662846730
FEUERBACH-TANGENTS
 CIRCUMPERP2
X(10)
X(5901)
FEUERBACH-TANGENTS
 EXCENTRAL
X(10)
X(5)
FEUERBACH-TANGENTS
 FEUERBACH
X(5)
X(5)
FEUERBACH-TANGENTS
 HEXYL
X(10)
X(550)
FEUERBACH-TANGENTS
 HUTSON INTOUCH
X(10)
(-S^2+3*SB*SC+2*(a-3*c)*SB*b+2*(a-3*b)*SC*c)/a
On lines: (1,550), (5,1697), (35,1387), (55,5901), (390,1482), (496,5119), (497,5690), (519,4536), (528,3884), (950,5844), (952,1898), (962,6147), (1317,5441), (1385,4342), (2098,4309), (3058,5697), (3295,3485), (3534,4308)
2.980513129092446
FEUERBACH-TANGENTS
 INTOUCH
X(10)
(-S^2+3*SB*SC+2*(a+c)*SB*b+2*(a+b)*SC*c)/a
Midpoint of: (942,4292)
On lines: (1,550), (3,7), (4,5708), (5,57), (11,79), (12,3336), (30,553), (36,3649), (40,4355), (46,495), (56,5901), (58,1086), (65,952), (84,5805), (140,226),         
 (354,1770), (355,3339), (382,938), (388,5690), (390,3296), (442,3218), (443,3927), (474,5905), (496,1836), (516,5045), (517,4298), (527,5044), (528,3881), (529,3754), (545,3159), (546,1210), (548,4114), (549,4654), (596,5846), (632,5219), (944,1159), (999,4295), (1385,3671), (1387,5563), (1407,5707), (1434,1565), (1478,5221), (1479,4860), (1482,3600), (1483,3340), (1595,1892), (1656,5435), (1657,3488), (2099,4317), (3295,3474), (3333,4312), (3526,5226), (3530,3982), (3534,4313), (3627,5722), (3628,3911), (3648,5284), (3678,5852), (3824,5745), (3851,5704), (3916,5249), (4303,5453), (4325,5425), (4757,5855), (4973,4999), (5326,5442), (5777,5843)
 0.400103889821620
FEUERBACH-TANGENTS
 MIDARC
X(10)
X(5901)

César E. Lozada
Feb. 20, 2015



Monday, February 16, 2015

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, 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^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 Zi:
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 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^9-4*(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 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*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 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 TAI and TAE their points of contact with the circumcircle. Build and name other circles cyclically.  Then:

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

·         ΔOAE OBE OCE 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 (TAI TAE), (TBI TBE), (TCI TCE) concur at X(25)

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

·         ΔTAETBETCE 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