ADGEOM 4864 * ADGEOM 4898 * ADGEOM 4901

#4864

Dear geometers,

 

I have a conjecture with Equal detour point X(176).

 

Let ABC be a triangle.

 

Let P0=X(176) of ABC.

 

Let Pa1=X(176) of P0BC.

 

Let Pa2=X(176) of Pa1BC.

 

....

 

Let Pan=X(176) of Pa(n-1)BC.

 

Define similarly, the points Pbn and Pcn.

 

Then triangle ABC and PanPbnPcn are perspective?

 

Best regards,

Tran Quang Hung.

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#4898

 

I will keep your notation P₀ = X(176)

The perspector (ABC, Pa1Pb1Pc1) = P₁ = X(482)

Now I will add another conjecture to your configuration:

If the perspector (ABC, PanPbnPcn) = P_n then

P_n = { X(7), P_n-1}-harmonic conjugate of P_n-2, for n>=2

= (a-b+c)*(a+b-c)*((n+1)*S+a*(-a+b+c)) : :  (barys)

= on the Soddy line X(1)X(7)

A short sequence of Pn:

 

P₂ = {X(7), X(482)}-HARMONIC CONJUGATE OF X(176)

= (a+b-c)*(a-b+c)*(3*S+a*(-a+b+c)) : : (barys)

= on lines: {1, 7}, {226, 3591}, {8965, 17092}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 176, 175), (7, 482, 176), (7, 17802, 1374), (7, 17805, 481), (175, 17804, 176), (176, 482, 17804), (481, 482, 1371), (481, 1371, 17805), (482, 1373, 7), (1371, 17805, 176), (1374, 17802, 17801), (17801, 17802, 175)

= [ 0.9611927715408633, 0.9994235592967669, 2.5051284309139110 ]

 

P₃ = X(1373) = {X(7), P₂}-HARMONIC CONJUGATE OF X(482)

 

P₄ = {X(7), X(1373)}-HARMONIC CONJUGATE OF P₂

= (a+b-c)*(a-b+c)*(5*S+a*(-a+b+c)) : : (barys)

= on the line {1, 7}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 17804, 176), (7, 482, 175), (7, 17804, 17801), (7, 17805, 1374), (175, 482, 176), (176, 17801, 1), (481, 482, 17806), (482, 1374, 17805), (1374, 17805, 175)

= [ 0.8875351968257142, 0.9296281795105022, 2.5874441129421630 ]

 

P₅ = {X(7), P₄}-HARMONIC CONJUGATE OF X(1373)

= (a+b-c)*(a-b+c)*(6*S+a*(-a+b+c)) : : (barys)

= on lines: {1, 7}, {226, 10194}, {553, 5393}, {3982, 13389}, {4114, 13388}, {4654, 5405}, {5589, 7613}

= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 176, 1374), (7, 482, 481), (7, 1373, 482), (17802, 17804, 17806)

= [ 0.8667352026601208, 0.9099188217431049, 2.6106890502421740 ]

 

César Lozada

PD: Do you have an easy method for finding X(176) in a given triangle?

Note: It seems that a sequence of perspective triangles is found if X(176) is instead of X(1).

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#4801

Dear Cezar,

 

Your formula

P_n = { X(7), P_n-1}-harmonic conjugate of P_n-2, for n>=2

= (a-b+c)*(a+b-c)*((n+1)*S+a*(-a+b+c)) : :  (barys)

 

is true also if we designate the point X(1) as P_-1

because X(482) = P₁ = { X(7), P₀}-harmonic conjugate of  P_-1=X(1).

Another conjecture is :

If  I_a0 is the incenter of triangle P₀BC then the line I_a0P_a1 passes through A

and

 

If Ia_n-1 is the incenter of triangle Pa_n-1BC then the line Ia_n-1Pa_n passes through Pa_n-2

Best regards

Nikos Dergiades