ADGEOM 4589 * ADGEOM 4613

#4589

 

Dear geometers,

 

Let ABC be a triangle and P,Q are isogonal conjugate points.

 

Ab, Ac are isogonal conjugate of B, C wrt triangle APQ.

 

Define similarly, the points Bc, Ba, Ca, Cb.

 

Then six points Ab, Ac, Bc, Ba, Ca, Cb lie on a conic.

 

When is it a circle?

 

Best regards,

Tran Quang Hung.

 

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#4613

 

Dear Tran Quang Hung,

 

I think the locus of points P such that the conic is a circle is complicated.  I found several such points on the Euler line, none are in ETC.

 

I did find several interesting observations from this configuration:

 

When P,Q are the Brocard points, PU(1):

The center is the trilinear pole of the line through X(351) parallel to the trilinear polar of X(351).

On lines {3,9217}, {6,694}, {32,249}, {39,512}, {543,598}, {574,805}, {733,12074}, {2086,3229} et al.

Barycentrics: a^2 (2 a^2 - b^2 - c^2) / (a^4 - b^2 c^2) : :

(6,9,13) values: (-0.765454011614075, 7.206375474085383, -0.995078225560544).

 

The perspector of the conic lies on lines {262,6036}, {523,3629}, {576,2065} et al

Barycentrics: a^2 (a^2 b^2 + a^2 c^2 - b^4 - c^4) / (a^4 + 2 b^4 + 2 c^4 - b^2 c^2 - 2 a^2 b^2 - 2 a^2 c^2) : :

(6,9,13) values:(-0.177721420347812, -0.199750198006888, 3.860978351457369).

 

The centroid of AbAcBcBaCaCb is X(10567).

 

The inverse of X(39) in the conic is X(882).

 

 

When P,Q are X(2), X(6):

The center of the conic is X(6).  The conic passes through X(2) and its antipode, X(1992) which are vertices of the conic.

 

The perspector of the conic has barycentrics: (5 a^2 - b^2 - c^2) / (14 a^4 + 2 b^4 + 2 c^4 - 20 a^2 b^2 - 20 a^2 c^2 + 13 b^2 c^2) : :

(6,9,13) values:(-0.654815572698841, -1.061665363990563, 4.677886152069619).

 

 

When P,Q are X(3), X(4), the conic is the Yff hyperbola, with center X(381) and perspector X(13481).

 

The centroid of AbAcBcBaCaCb is X(13448).

 

 

When P,Q are X(13), X(15):

The conic is degenerate, 2 lines intersecting at X(13).

 

The centroid of AbAcBcBaCaCb lies on lines {13,15}, {1989,9140}.

 

 

When P,Q are X(14), X(16):

The conic is degenerate, 2 lines intersecting at X(14).

 

The centroid of AbAcBcBaCaCb lies on lines {14,16}, {1989,9140}.

 

 

Best regards,

Randy Hutson