ADGEOM 2046 * ADGEOM 2047 * ANOPOLIS 1463 * ANOPOLIS 1464

#2046

 

Dear geometers

Let ABC be a triangle.

Incircle touches BC,CA,AB at A',B',C'

A-Excircle (Ia) touches BC,CA,AB at Aa,Ab,Ac

B-Excircle (Ib) touches CA,AB,BC at Bb,Bc,Ba

C-Excircle (Ic) touches AB,BC,CA at Cc,Ca,Cb

Then circumcircle (wa) of triangle A'BcCb passes through Feuerbach point Fa on (Ia). Let (wa) cuts (Ia) again at A''. Similar we have B'',C''. Then AA'',BB'',CC'' are concurrent.

Best regards,
Tran Quang Hung.

 

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#2047

Dear Tran,

Very nice!  AA", BB". CC" concur in the isotomic conjugate of the polar conjugate of X(2197), with trilinears (cos^2 A)(1 + cos(B - C)) : :, on lines {3,3173} {10,12} {55,581} {56,219} {71,73} (at least).  ETC search = 2.690012319586278.

A related point can be obtained by letting (wa) cut the incircle again at A*, and cyclically for B*, C*.  Then AA*, BB*, CC* concur in the polar conjugate of X(219) = isogonal conjugate of X(6056) = isotomic conjugate of X(1259), with trilinears (csc^2 2A)(1 - cos A) : :, barycentrics csc 2A (1 - sec A) : : == csc 2A - 2 csc^2 2A : : == sec^2 A tan A/2 : : == b^2c^2/[(b + c - a)(b^2 + c^2 - a^2)^2] : :, on lines {7,286} {92,226} {158,273} {264,1441} (at least).  ETC search = 0.246657534644698.

This last point was also found by Antreas Hatzipolakis in Anopolis #1463, 5/6/2014:
Let Ia, Ib, Ic be the excenters, and H the orthocenter, X(4).  The nine-point circle of AHIa intersects the A-excircle at the A-ex-Feuerbach point and another point.  Call this second point A', and define B', C' cyclically.  The lines AA', BB', CC' concur in this point.

Best regards,
Randy Hutson

 

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Anopolis #1463

Let ABC be a triangle with excenters Ia,Ib,Ic.

The NPC of  AHIa interscts the excircle (Ia) at A' other than the

Feuerbach point Fa.

The  NPC of  BHIb intersects the excircle (Ib) at B' other than the

Feuerbach point Fb,

The  NPC of  CHIc intersects the excircle (Ic) at C' other than the

Feuerbach point Fc.

Are the triangles ABC, A'B'C' perspective?

In any case, has the triangle A'B'C' any interesting properties?

Antreas P. Hatzipolakis

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#Anopolis 1464

Hi Antreas,

Very nice!  I have been away from the list for awhile, trying to finish up a project on polar conjugates, when I happened to see your post, and decided to take a stab at it.  It turns out A'B'C' and ABC are indeed perspective, and the perspector is the polar conjugate of X(219) = isotomic conjugate of X(1259), on lines 7,286 92,226 264,1441 (at least).  It has trilinears (csc^2 2A)(1 - cos A) : :, barycentrics (csc 2A)(1 - sec A) : :, and ETC search value 0.246657534644698.  It is also the trilinear pole of the line through the polar conjugate of X(100) and the isotomic conjugate of X(645).

Best regards,
Randy Hutson