ADGEOM 4850 * ADGEOM 4851 * ADGEOM 4853 * ADGEOM 4855 * ADGEOM 4857

#4850

Dear geometers,

 

Let ABC be a triangle.

 

Contact triangle A'B'C' and Nagel point Na.

 

Let La,Lb,Lc be the symmedian points triangles NaB'C', NbC'A' and NcA'B'.

 

Then ABC and LaLbLc are perspective.

 

Which is the perpsector?

 

Best regards,

Tran Quang Hung.

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#4851

[Tran Quang Hung]:

Let ABC be a triangle.

Contact triangle A'B'C' and Nagel point Na.

Let La,Lb,Lc be the symmedian points triangles NaB'C', NbC'A' and
NcA'B'.

Then ABC and LaLbLc are perspective.

Which is the perpsector?

*** ABC and LaLbLc are perspective at
W = (4 r^2+16 r R-s^2) X(1) - 12 r^2 X(2)

W = (a^3-2 a^2 (b+c)-4 (b-c)^2 (b+c)+a (9 b^2-14 b c+9 c^2) : -4
a^3+b^3-2 b^2 c+9 b c^2-4 c^3+a^2 (9 b+4 c)-2 a (b^2+7 b c-2 c^2) : -4
a^3-4 b^3+9 b^2 c-2 b c^2+c^3+a^2 (4 b+9 c)+2 a (2 b^2-7 b c-c^2))

The (6 - 9 - 13) - search numbers of W are
(3.45104654014144, 1.80769104483599, 0.796395355417256),
which are the same coordinates that the center
X(4691) = (-2 a+5 b+5 c : 5 a-2 b+5 c : 5 a+5 b-2 c).

Angel Montesdeoca

 

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#4853

 

Hi Angel,

 

Interesting.  So now we have a second pair of points (besides X(3635) and X(15519)) that are exactly the same point when the reference triangle is (6,9,13).  It seems that a second set of search tables for another reference triangle, preferably acute (e.g. (7,9,11)), would be helpful in cases like this.

 

Best regards,

Randy Hutson

 

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#4855

 

Hi Angel, Randy and Mr Hung,

 

W = complement of X(15519)

= a^3-2*(b+c)*a^2+(9*b^2-14*b*c+9*c^2)*a-4*(b^2-c^2)*(b-c) : : (barys)

= (4 r^2+16 r R-s^2) X(1) - 12 r^2 X(2)

= on lines: {1, 2}, {165, 3021}, {1997, 4929}, {3756, 8056}, {4862, 5274}, {7963, 12625}

= complement of X(15519)

= [3.451046540141442691497607, 1.807691044835993790784461, 0.7963953554172560057302172]

 

César Lozada

 

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#4857

 

The point W lies also on line {1699, 4902}

Ercole Suppa