ADGEOM 3963 * ADGEOM 3966 * ADGEOM 3970

#3963

Dear geometers,

 

Let ABC be a triangle with two Brocard points W1, W2.

 

Let A1B1C1 be cevian triangle of W1.

 

Let A2B2C2 be cevian triangle of W2.

 

Then the perpendicular bisectors of A1A2, B1B2, C1C2 are concurrent.

 

Which is this point ?

 

Best regards,

Tran Quang Hung.

 

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#3966

 

[Tran Quang Hung]:

    Let ABC be a triangle with two Brocard points W1, W2.

    Let A1B1C1 be cevian triangle of W1.

    Let A2B2C2 be cevian triangle of W2.

    Then the perpendicular bisectors of A1A2, B1B2, C1C2 are concurrent.

    Which is this point ?
   
   
***** GENERALIZATION:  (See  http://bernard.gibert.pagesperso-orange.fr/Exemples/k659.html)

Consider a point P = u : v : w and its two Brocardians P1 = 1/w : 1/u : 1/v, P2 = 1/v : 1/w : 1/u. See Tucker cubics for more details.

The cevian triangles of P1, P2 are (P1a, P1b, P1c) and (P2a, P2b, P2c). The perpendicular bisectors of the three segments P1x-P2x are concurrent (at Q) if and only if P lies on K659. (Angel Montesdeoca, private message, 2014-01-24)

P = X(2)  ---> Q = X(3)

P = X(6) --->  Q =  X(3)X(695) /\ X(4)X(83) =
                    (a^2 (-a^6 (b^2+c^2)+3 a^2 b^2 c^2 (b^2+c^2)+a^4 (b^4+c^4)+b^2 c^2 (b^4+c^4)):...:...),
                    with  (6 - 9 - 13) - search numbers (4.67367749959898, 4.53236421976679, -1.65420805466908)

P = X(76)  ---> Q = X(3)X(695) /\ X(20)X(1352) =
      (a^2 (a^6 (b^2+c^2)+a^4 (b^4+c^4)-b^2 c^2 (b^4+4 b^2 c^2+c^4)-a^2 (2 b^6+b^4 c^2+b^2 c^4+2 c^6)):...:...),
       with  (6 - 9 - 13) - search numbers (8.89104828879372, 7.25753763369601, -5.48734539778688)

P = X(194) ---> Q = X(1)X(7153) /\ X(3)X(3229) =
          (a^2 (a^4 (b^4-4 b^2 c^2+c^4)+b^2 c^2 (b^4-4 b^2 c^2+c^4)-a^2 (b^6-4 b^4 c^2-4 b^2 c^4+c^6)):...:...),
           with  (6 - 9 - 13) - search numbers (0.183016621174312, 3.81829238807004, 0.912762080393899)


P = X(2998)  ---> Q = X(6310)

Angel Montesdeoca

 

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#3970

 

 

Thank you very much dear Mr Angel Montesdeoca,

 

I have seen the similar problem as following

 

Let ABC be a triangle with two Brocard points W1, W2.

 

Let A1B1C1 be cyclocevian triangle of W1.

 

Let A2B2C2 be cyclocevian triangle of W2.

 

Then the perpendicular bisectors of A1A2, B1B2, C1C2 are concurrent.

 

Is this problem in your generalization ?

 

Best regards,

Tran Quang Hung.

ADGEOM 3866 *