ADGEOM 1984 * ADGEOM 2002

#1984

Dear Dr. Francisco Javier, Mister Dergiades, and all Friends

 

I proposed problem as follows:

 

I an very thank to You. I proposed another new circles:

 

Let ABC be a triangle, V_AV_BV_C be the outer or inner Vecten  triangle of ABC. Let M_a,M_b,M_c be three midpoint of

BC,CA,AB respectively. Let  O_a,O_b,O_c be center of three circle (V_BV_CM_a), (V_CV_AM_b), (V_BV_AM_c)

respectively. Let O_{BC}, O_{CA}, O_{AB} be another intersection point of three circle (V_BV_CM_a), (V_CV_AM_b), 

(V_BV_AM_c) respectively. O_{BC}, O_{CA}, O_{AB} , O_A,O_B,O_C lie on a circle(I have not mathematica sofware)

 

Could you help me answerthat the center of two pair circle are X(?) in Kimberling ETC.

 

Please see the figure attachments.

 

 

Best regards

Sincerely

 Dao Thanh Oai

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#1987

 

The centers are the points with first coordinate

(a^2 - b^2 - c^2) (2 a^4 - a^2 b^2 + b^4 - a^2 c^2 - 2 b^2 c^2 + c^4) (+/-) 2 S (6 a^4 - 7 a^2 b^2 + 7 b^4 - 7 a^2 c^2 - 10 b^2 c^2 + 7 c^4)

where

 S = twice the area of ABC
+/- means a + for outer vecten point and a - for iner vecten points.

Best regards,
Francisco Javier Garcia Capitan

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#1991

 

Dear Dao:

Two remarks:

1) The line joining the two centers goes through the points X575, X3564, X3589, X3628, X5449
2) The two centers are harmonic conjugates with respect to the pair  X3589, X3628.

Best regards,
Francisco Javier Garcia Capitan

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#1992

 

Dear Dao:

3) The points X2501, X3566, X5203 lie on the radical axis of the two circles.

Best regards,
Francisco Javier Garcia Capitan

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#1999

 

Dear Dao,

You found very nice circles!

If we call Vecten configuration the 1st one, there are the 2nd, 3rd ones... See FG200637 "Square Wreaths around Hexagons".

It would be very interesting if, in addition to the1st inner and outer circles, there were the 2nd, the 3rd ones...

 

P. S. There are an infinite number of trilinear points concerning Vecten configuration and its extension.

See Hyacinthos message #22011.

Best regards,

Seiichi Kirikami

 

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#2001

 

Dear Geometer,

 

I am thank to Dr. Garcia Capitan, Dr. Moses, and Mister. Seiichi and all Friends

 

In this con figuration I see that:

 

AO_A,BO_B,CO_C are concurrent. Where O_A,O_B,O_C are center of three circle (V_BV_CM_a),(V_CV_AM_b), (V_AV_BM_c). 

 

and 

 

AI_{BC}, BI_{CA}, CI_{AB} are concurrent. Where I_{BC}, I_{CA}, I_{AB} are intersection(again) of three circles (V_BV_CM_a),(V_CV_AM_b), (V_AV_BM_c). 

 

Best regards

Sincerely

Dao Thanh Oai

 

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#2002

 

In the figure:

 

Then: AO_{BC},BO_{CA},CO_AB} are concurrent, and AO_A,BO_B,CO_C are concurrent. How are number of these points in Kimberling Center?

 

  Dao Thanh Oai