ADGEOM 3989 * ADGEOM 3990 * ADGEOM 3992 * ADGEOM 3997

#3989

Dear geometers,

 

Let ABC be a triangle with circumcircle (O).

 

Excenters are Ia,Ib,Ic.

 

AIa,BIb,CIc meet (O) again at A',B',C'.

 

Lester circle of triangle IaBC meet (O) again at A''.

 

Similarly, we have B'',C''.

 

Then triangle ABC and A''B''C'' are perspective, which is the perspector ?

 

Best regards,

Tran Quang Hung.

 

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#3990

 

Dear Tran Quang Hung

The perspector of the triangles ABC and A''B''C'' is

W = (a (a-b) (a-c) (a^10 -a^9 (b+c) +a^8 (-3 b^2+5 b c-3 c^2) +a^7 (4 b^3-2 b^2 c-2 b c^2+4 c^3) +a^6 (2 b^4-5 b^3 c+5 b^2 c^2-5 b c^3+2 c^4) -a^5 (6 b^5-6 b^4 c+b^3 c^2+b^2 c^3-6 b c^4+6 c^5) +a^4 (2 b^6-3 b^5 c-4 b^4 c^2+11 b^3 c^3-4 b^2 c^4-3 b c^5+2 c^6) +a^3 (b-c)^2 (4 b^5+6 b^4 c+7 b^3 c^2+7 b^2 c^3+6 b c^4+4 c^5) -a^2 (b^2-c^2)^2 (3 b^4-b^3 c+b^2 c^2-b c^3+3 c^4) -a (b-c)^4 (b+c)^5 +(b-c)^4 (b+c)^6) : ... :  ...),
                           
with  (6 - 9 - 13) - search numbers   (19.1358883812807, -29.4610758527141, 15.2048454347338).
W lies on line  X(37)X(11075).

Best regards,
Angel Montesdeoca

 

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#3992

 

Thank you very much Mr Angel Montesdeoca, I have seen the similar problem with Lester circle as following

 

Let ABC be a triangle inscribed in (O) with incenter I.

 

A'B'C' is circumcevian triangle of I.

 

Lester circle of triangle IBC,ICA,IAB meet (O) again at A'',B'',C''.

 

Then AA'',BB'',CC'' meets BC,CA,AB respectively at three collinear point on a line. Which is this line ?

 

Best regards,

Tran Quang Hung.

 

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#3997

 

Dear Tran Quang Hung


AA'',BB'',CC'' meets BC,CA,AB respectively at three collinear point on a line

b c (b+c) (a^8+a^6 (-4 b^2+3 b c-4 c^2)+(b-c)^2 (b+c)^4 (b^2+b c+c^2)+a^4 (6 b^4-3 b^3 c+b^2 c^2-3 b c^3+6 c^4)+a^2 (-4 b^6-3 b^5 c+b^4 c^2+9 b^3 c^3+b^2 c^4-3 b c^5-4 c^6)) x + .... =0,

 trilinear polar of

Z =  3R(2r+3R) (r(r+2R)+s^2) X(79) - 2 ((r+3R)^2-3s^2) (r(r+R)-s^2) X(81)

Barycentric coordinates:

Z = (a (a^3+a^2 (b+c)-(b-c)^2 (b+c)-a (b^2+b c+c^2))/((b+c)  (-a^2+b^2+b c+c^2)) : .... : ...),

with (6 - 9 - 13) - search numbers: (-0.00980661461614026, -0.0145626594472695, 3.65527245288609).

Best regards,
Angel Montesdeoca

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