ADGEOM 4796 * ADGEOM 4798 * ADGEOM 4801 * ADGEOM 4808

#4796

Dear geometers,

I see these two points have this property.

 

A is X(74) of triangle X(74)BC.

 

A is also X(1138) of triangle X(1138)BC.

 

Are these property easy seen?

 

Best regards,

Tran Quang Hung.

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#4798

Dear Mr. Tran Quang Hung,

Yes, they are easily seen with the following properties:

X(74) is an intersection of the circumcircle and the Jerabek hyperbola

=> X(74) is an intersection of the circle ABCX(74) and the hyperbola ABCX(74)X(3).

Hence ABCX(74)X(3) is also the Jerabek hyperbola with respect to the triangles X(74)BC, X(74)CA, X(74)AB.

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The Euler lines of the triangle ABC, X(1138)BC, X(1138)CA, X(1138)AB are parallel.

 

Sincerely

Ngo Quang Duong

 

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#4801

 Another property of points  X(74) and X(1138):

Let A' be the intersection point of the Euler lines of triangles X(74)BC and X(1138)BC,
and define B', C' cyclically.
Then triangle ABC and A'B'C' are perspective, with perpsector

W = (a^2-b^2-c^2) (2 a^4-a^2 (b^2+c^2)-(b^2-c^2)^2)/(a^8-4 a^6 (b^2+c^2)+a^4 (6 b^4+b^2 c^2+6 c^4)+a^2 (-4 b^6+b^4 c^2+b^2 c^4-4 c^6)+(b^2-c^2)^2 (b^4+4 b^2 c^2+c^4)) : ... : ....

with  (6 - 9 - 13) - search numbers:  (6.26250855866268, 3.50369184105253, -1.67535689666554).

W lies on lines X(i)X(j) for these {i, j}: {30,146}, {74,18317}, {265,14919}, {1294,14677}, {1494,10264}, {1511,3163}, {6699,8552}, {10272,14920}, {16163,19223}.

Angel Montesdeoca

 

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#4808

This brings up a question I have often wondered about.  Perhaps it has already been discussed.  Are there any other points besides X(4), X(74) and X(1138) that form 'X-centric systems'?  That is, every point in the set {A,B,C,X} is 'X' of the remaining three points.

 

Best regards,

Randy Hutson