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)*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

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