ADGEOM 3904

 

[Tran Quang Hung]:

Dear geometers,

 

Let ABC be a triangle and A'B'C' is cevian triangle of X(31).

A'',B'',C'' are X(31) of the triangle AB'C',BC'A',CA'B'.

Then AA'',BB'',CC'' are concurrent at a point.

 Which is this point.

 More general.

 Let ABC be a triangle. We define point Px(a^x,b^x,c^x) in barycentric coordinate with x is a real number. 

 Let P be Px point of triangle ABC and A'B'C' is cevian triangle of P.

 A'',B'',C'' are the point Px of the triangle AB'C',BC'A',CA'B'.

 Then AA'',BB'',CC'' are concurrent at a point.

 Which is the locus when x change ?

 

Best regards,

Tran Quang Hung.

 

[César Lozada]:

 

Dear Mr. Hung:

 

Hope you don´t mind that my answer uses trilinear coordinates:

 

If P(k) = a^k : b^k : c^k   (trilinears), the perspector Q(k) of A” and A is

Q(k): 1/((a^k*(b^(k+1)+c^(k+1)))^k) : :  (trilinears)

 

In the following list, (k, I) means that P(k) = X(I)

(-6, 1928), (-5, 1502), (-4, 561), (-3, 76), (-2, 75), (-1, 2), (0, 1), (1, 6), (2, 31), (3, 32), (4, 560), (5, 1501), (6, 1917), (7, 9233)

 

In this one, (k, I) means that Q(k)=X(I):

(-2, 1089), (-1, 2), (0, 1), (1, 83)

 

Q(2) = 1/(a^4*(b^3+c^3)^2) : : (tri.)

= on lines: {}

= [ 3.316794374500031, 0.96335756403065, 1.442896457040066 ]

 

Q(-3) = (b^2+c^2)^3/a^3 : : (tri.)

= On lines: {305,7855}, {3266,5041}

= [ 6.226367119072926, 1.01719098835822, 0.062747435010413 ]

 

César Lozada