HYACINTHOS 26620

Let ABC a triangle with incircle {i} and inradius r. The tangent line to {i} parallel to BC cuts AB, AC at A_b, A_c, respectively. Denote {i_a} and r_a the incircle and inradius of AA_bA_c and similarly {i_b}, r_b and {i_c}, r_c. Then  r_a + r_b  + r_c = r. (http://euclidsmuse.com/ members/polymnia/apps/app/1042 )

 

Notes:

 r_a = r*(-a+b+c)/(a+b+c). The center of {i_a} is I_a = (a+3*b+3*c)/(-a+b+c) : 1 : 1 (trilinears)

 

The radical center of {i_a}, {i_b}, {i_c} is:

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

= On lines: {165, 1202}, {354, 10481}, {991, 995}, {1362, 5919}, {5049, 5542}

= [ 2.899589408548284, 2.97796482447824, 0.240724491784853 ]

 

The circle externally tangent to {i_a}, {i_b}, {i_c} is also tangent to the incircle {i} at X(1362). This circle has squared radius ρ² =  r^2*(s^4+(4*R+r)^2*((4*R+r)^ 2+2*s^2))/(16*s^4) and center:

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

= (4*R*s^2-SW*(4*R+r))*X(3)+SW*( 4*R+r)*X(6)

= On lines: {1, 1362}, {3, 6}, {51, 11349}, {942, 10481}, {946, 971}, {3333, 4334}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (991, 4253, 3), (6479, 8410, 1030)

= [ 2.317743959417002, 2.49751982706278, 0.841884312671531 ]

 

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Let A₀ be the touchpoint of {i_a} and A_bA_c and similarly B₀, C₀ (A₀ = 4*a/(-a+b+c) : (a+b-c)/b : (a-b+c)/c, trilinears). A₀B₀C₀ is perspective with perspector X(7) to these triangles: ABC, Conway, 2^nd Conway, Honsberger, Hutson-extouch and intouch. It is also perspective to the following triangles at the given points:

 

·         extangents: (a^5-3*(b+c)*a^4+2*(b^2+4*b*c+ c^2)*a^3+2*(b+c)*(b^2-10*b*c+ c^2)*a^2-(b+c)*(b^2+6*b*c+c^2) *(3*(b+c)*a-(b-c)^2))*(a-b+c)* (a+b-c) : : (trilinears)

= On lines: {57, 71}, {65, 5223}

= [ 0.602097057594823, 0.76200827248238, 2.835229343606659 ]

 

·         intangents: (-a+b+c)*(a^6-3*(b-c)^2*a^4-16*(b^2-c^2)*(b-c)*b*c*a+(b^2+6*b*c+c^2)*(b-c)^2*(3*a^2-(b-c)^2)) : : (trilinears)

= On lines: {1, 7955}, {33, 8917}, {282, 522}, {354, 4328}, {3938, 4319}, {10703, 11224}

= [ 3.645387210301032, 3.98599237387348, -0.801355104759275 ]

 

·         inverse-in-incircle: (b+c)*a^3-(3*b^2-2*b*c+3*c^2)* a^2+(b+c)*(3*b^2-4*b*c+3*c^2)* a-(b^2+c^2)*(b-c)^2 : : (trilinears)

= 3*X(354)-X(1122)

= On lines: {1, 6}, {65, 13572}, {354, 1122}, {938, 5015}, {942, 4307}, {971, 4310}, {982, 1742}, {2835, 12016}, {2887, 11019}, {3008, 3059}, {3663, 14100}, {3672, 7671}, {3677, 10391}, {3945, 11025}, {3976, 4334}, {4388, 10580}, {9309, 12915}, {9844, 13161}

= [ 1.816586701363288, 1.72669375119273, 1.606836484298661 ]

 

·         Mandart-excircles: (a+b-c)*(a-b+c)*((b+c)*a^4-4* b*c*a^3-2*(b+c)*(b^2-3*b*c+c^ 2)*a^2+(b^4-c^4)*(b-c)) : : (trilinears)

= On lines: {6, 57}, {65, 516}, {77, 4254}, {169, 6180}, {198, 4341}, {241, 573}, {1020, 1108}, {1214, 2269}, {1400, 9502}, {1828, 1876}, {2262, 3668}, {5120, 7013}, {8270, 10387}

= [ 0.343362589212971, 0.55539360777121, 3.097686173813709 ]

 

·         tangential: (a^6-2*(b+c)*a^5+(b^2+4*b*c+c^ 2)*a^4-2*(b+c)*b*c*a^3-(b^4+c^ 4-2*(b^2-13*b*c+c^2)*b*c)*a^2+ 2*(b^3+c^3)*(b-c)^2*a-(b^4-10* b^2*c^2+c^4)*(b-c)^2)*a : : (trilinears)

= On lines: {527, 4428}, {910, 2285}, {1279, 3304}, {1486, 4644}, {3340, 3827}

= [ 0.922769388981600, 1.27857016559205, 2.329606956967607 ]

 

César Lozada