ADGEOM 3920 * ADGEOM 3925 * ADGEOM 3926

#3920

Dear geometers,

 

Let ABC be a triangle with NPC center N.

 

Na,Nb,Nc are NPC centers of triangles NBC,NCA,NAB.

 

A'B'C' is anticevian triangle of N wrt triangle NaNbNc.

 

Then N is incenter of triangle A'B'C'.

 

So Euler line of ABC is which line of A'B'C' ?

 

Best regards,

Tran Quang Hung.

 

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#3925

 

[Tran Quang Hung]:

Dear geometers,

 

Inircle (N) of  A'B'C' touches B'C',C'A',A'B' at A'',B'',C'' then triangle ABC and A''B''C'' are similar and orthologic. The orthology center lies on their circumcircles.

 

Which is the center of similar transformation which transform A''B''C'' to ABC ?

 

Best regards,

Tran Quang Hung.

 

 

[César Lozada]:

>>triangles ABC and A''B''C'' are similar and orthologic

 

Both affirmations only if ABC is acute

 

For ABC acute:

 

Orthologic centers:

Q(A->A”) =  X(1141)

 

Q(A”->A) = complement of X(13630)

= a^2*((b^2+c^2)*a^2-(b^2-c^2)^2)*(a^4-2*(b^2+c^2)*a^2+5*b^2*c^2+c^4+b^4) : : (barycentrics)

= 3*X(2)+X(5876) = 9*X(3)-X(12279) = 7*X(3)+X(12290) = 5*X(4)+3*X(13340) = 7*X(5)-3*X(51) = 5*X(5)-X(52) = 3*X(5)-X(143) = 3*X(5)+X(5562) = 15*X(51)-7*X(52) = 9*X(51)-7*X(143) = 9*X(51)+7*X(5562) = 5*X(10627)-3*X(13340) = 7*X(12279)+9*X(12290)

= On lines: {2, 5876}, {3, 6030}, {4, 10627}, {5, 51}, {30, 5447}, {110, 10610}, {140, 5663}, {156, 7514}, {185, 632}, {381, 6101}, {382, 7999}, {389, 547}, {511, 3850}, {546, 1216}, {548, 3819}, {549, 12162}, {568, 5056}, {631, 13491}, {1493, 13434}, {1656, 6102}, {1657, 7998}, {2063, 9818}, {2979, 3843}, {3060, 5072}, {3090, 5946}, {3091, 10263}, {3526, 12111}, {3530, 6000}, {3534, 11439}, {3545, 6243}, {3567, 5079}, {3627, 3917}, {3628, 10219}, {3845, 10625}, {3851, 11412}, {4550, 11250}, {5054, 6241}, {5055, 5889}, {5066, 5446}, {5070, 5890}, {5609, 7550}, {5650, 10575}, {5943, 12046}, {7564, 10516}, {7723, 11561}, {8703, 11381}, {10024, 12358}, {10110, 12811}, {12103, 13474}, {12134, 13470}

= midpoint of X(i) and X(j) for these {i,j}: {4, 10627}, {5, 11591}, {140, 5907}, {143, 5562}, {546, 1216}, {5876, 13630}, {7723, 11561}, {11412, 13421}, {11459, 13363}, {12103, 13474}, {12134, 13470}

= reflection of X(i) in X(j) for these (i,j): (546, 11017), (548, 11592), (10095, 5), (10110, 12811), (12006, 3628)

= complement of X(13630)

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (2, 5876, 13630), (5, 52, 13364), (5, 5562, 143), (5, 5891, 11591), (143, 11591, 5562), (381, 11444, 6101), (548, 3819, 11592), (1656, 6102, 13363), (1656, 11459, 6102), (5907, 10170, 140), (5943, 12812, 12046)

= [ 0.877056360830461, -0.43956653222597, 3.540184529988299 ] (calculated for 6-9-13 triangle, although A”B”C” and ABC are not orthologic for it)

 

Center of inverse similitude (ABC, A”B”C”)

Si = 4*(2*cos(2*A)-7)*cos(B-C)-2*(3*cos(A)-cos(3*A))*cos(2*(B-C))-7*cos(3*A)+cos(5*A)-5*cos(A) :: (trilinears)

= [ 0.698634769708212, -0.61918455028304, 3.746883892238080 ]

 

César Lozada

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#3926

 

The orthology center of ABC wrt A''B''C'' is X(1141).

The orthology center of A''B''C'' wrt ABC is  7 X(5) - 3 X(51),

with (6 - 9 - 13) - search numbers (0.877056360830461, -0.439566532225970, 3.54018452998830).

The center of similar transformation which transform A''B''C'' to ABC  is

W = ( a^2 (a^14-5 a^12 (b^2+c^2)+a^10 (11 b^4+8 b^2 c^2+11 c^4)+a^8 (-15 b^6+b^4 c^2+b^2 c^4-15 c^6)-(b^2-c^2)^4 (b^6-3 b^4 c^2-3 b^2 c^4+c^6)+a^2 (b^2-c^2)^2 (5 b^8-6 b^6 c^2-34 b^4 c^4-6 b^2 c^6+5 c^8)+a^6 (15 b^8-8 b^6 c^2-13 b^4 c^4-8 b^2 c^6+15 c^8)+a^4 (-11 b^10+13 b^8 c^2+33 b^6 c^4+33 b^4 c^6+13 b^2 c^8-11 c^10)) : ... : ...).

with (6 - 9 - 13) - search numbers (0.698634769708212, -0.619184550283042, 3.74688389223808).

Angel Montesdeoca