ADGEOM 1728 * ADGEOM 1730 * ADGEOM 1733

#1728

[Seiichi Kirikami]:

Dear friends, dear Randy,

Let ABC be a triangle with its circumcircle k.

The trisector of angle A lying near AC intersects k at Ac.

The trisector of angle A lying near AB intersects k at Ab.

The trisector of angle B lying near BA intersects k at Ba.

The trisector of angle B lying near BC intersects k at Bc.

The trisector of angle C lying near CB intersects k at Cb.

The trisector of angle C lying near CA intersects k at Ca.

Denote the intersection of BBc and CCb by D, the intersection of CCa and AAc by E, and the intersection of AAb and BBa by F. DEF is the 1^st Morley triangle.

Denote the intersection of AcCa and AbBa by D1, the intersection of BaAb and BcCb by E1, and the intersection of CbBc and CaAc by F1. D1E1F1 is the Roussel triangle.

Denote the intersection of AcBc and AbCb by D2, the intersection of BaCa and BcAc by E2, and the intersection of CbAb and CaBa by F2. D2E2F2 is the adjunct Roussel triangle.

The 1^st Morley triangle DEF and the adjunct Roussel triangle D2E2F2 are perspective and have the following concurrent point.

 

Trilinears={-2Cos[5B/3+5C/3] + 2Sqrt[3]Cos[B+C+Pi/6] + 2Sin[B/3-C-Pi/6] + 2

Sin[B/3+C/3-Pi/6] -  Sin[5B/3+C-Pi/6] - 2Sin[B+5C/3-Pi/6] – 2 Sin[B-C/3+Pi/6]:  :  }.

Its length=7.

Best regards,

Seiichi.

P. S. My computation showed that Roussel triangle D1E1F1 and adjunct Roussel triangle D2E2F2 were not perspective.

 

#1730

[Randy Hutson]:

Dear Seiichi,

Your construction of the adjunct Roussel triangle agrees with mine, but I do find that it is perspective to the Roussel triangle, and the perspector has ETC search value 2.427937089459433.  It is also perspective to the 1st Morley triangle at ETC search=1.499634729695783 (on lines 15,358 and 3280,3602).

I am not sure what you mean by length=7.

Best regards,
Randy Hutson

 

#1733

[Randy Hutson]:

Dear Seiichi and friends,

Another triangle can be constructed from this configuration:

Denote the intersection of BcBa and CaCb by D3, the intersection of CaCb and AbAc by E3 and the intersection of AbAc and BcBa by F3.  D3E3F3 is homothetic to ABC at trilinears cos A - cos(A/3) : : (Search=0.195460788980394, on lines 3,358 357,3279 3280,3602), and perspective to the Roussel triangle D1E1F1 and the adjunct Roussel triangle D2E2F2 at (search=2.427937089459433).

PS. Disregard my earlier question about length=7.  I see you are referring to the number of terms in the coordinates.

Best regards,
Randy Hutson.