Let A, B, C be three points that do not lie on a line...: 2015

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 ΔP_AP_BP_C be the triangle having vertices at the contact points of {Ap} and the excircles and ΔT_AT_BT_C 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 ΔP_AP_BP_C as APOLLONIUS-CONTACT-TRIANGLE of ΔABC, ΔT_AT_BT_C 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)SBb-(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-2s^2-bc)(SA+2s^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(2b^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((2a-s)S^2-SA(SWs-3abc-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(2SA(2S(R-r)+SWs)-(3a-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((2RS-sSW)SA+sS^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((7a-b-c)S^2+2(3a-s)SA^2-2(s-b)SASB-2(s-c)SASC) 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+sS^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)	(5S^2-7SBSC-2(a-c)SBb-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+3SBSC+2(a-3c)SBb+2(a-3b)SCc)/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+3SBSC+2(a+c)SBb+2(a+b)SCc)/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

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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, 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 ΔT_AET_BET_CE - TANGENTIAL triangle of ABC
-1.08919401759083	Exsimicenter of Γ_I, Γ_E
0.58032649700693	Reflection of X(11) in Z_I
1.61173080912606	Center of Γ_I= Z_I
3.30522235113698	Reflection of X(108) in Z_I
4.71391292078389	Perspector ΔO_AEO_BEO_CE - MEDIAL triangle of ABC
4.94891114159610	Center of Γ_E= Z_E
7.20389309624506	Reflection of X(105) in Z_E
7.25468716194700	Reflection of X(11) in Z_E

César Lozada

Feb 16, 2015

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