#3874
Dear geometers,
Let ABC be a triangle with three Feuerbach points Fa,Fb,Fc.
H is orthocenter.
Fha is H-Feuebrach point of triangle HBC.
Fhb is H-Feuebrach point of triangle HCA.
Fhc is H-Feuebrach point of triangle HAB.
FaFha meets BC at A'. Similarly, we have B',C'.
Then the lines AA',BB',CC' are concurrent. Which is this point ?
Best regards,
Tran Quang Hung.
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#3878
Dear Tran Quang Hung
The lines AA',BB',CC' are concurrent at
W = ( 2 a^6-a^5 (b+c)-a^4 (3 b^2+2 b c+3 c^2)+2 a^3 (b^3+b^2 c+b c^2+c^3)-a (b-c)^2 (b+c)^3+(b-c)^2 (b+c)^4+4 (-a^4+a^3 (b+c)-a^2 (b+c)^2-a (b-c)^2 (b+c)+2 (b^2-c^2)^2) r s : ... : ...).
W is the midpoint of X(176) and X(10405).
W lies on lines X(i)X(j) for these {i, j} : {1,1336}, {2,175}, {4,9}, {176,10405}, {189,13389}, {219,1378},{226,13459}, {278,3535}, {388,6204, {497,7347}, {637,1944}, {1123,1785}, {1146,3070}, .....
(6 - 9 - 13) - search numbers of W : (7.51986707668436, 5.97337161677889, -3.96545451894763).
Best regards,
Angel Montedeoca
The lines AA',BB',CC' are concurrent at
W = ( 2 a^6-a^5 (b+c)-a^4 (3 b^2+2 b c+3 c^2)+2 a^3 (b^3+b^2 c+b c^2+c^3)-a (b-c)^2 (b+c)^3+(b-c)^2 (b+c)^4+4 (-a^4+a^3 (b+c)-a^2 (b+c)^2-a (b-c)^2 (b+c)+2 (b^2-c^2)^2) r s : ... : ...).
W is the midpoint of X(176) and X(10405).
W lies on lines X(i)X(j) for these {i, j} : {1,1336}, {2,175}, {4,9}, {176,10405}, {189,13389}, {219,1378},{226,13459}, {278,3535}, {388,6204, {497,7347}, {637,1944}, {1123,1785}, {1146,3070}, .....
(6 - 9 - 13) - search numbers of W : (7.51986707668436, 5.97337161677889, -3.96545451894763).
Best regards,
Angel Montedeoca
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