ADGEOM 1256 * ADGEOM 1259

 

#1256

Dear Mister Clack Kimberling, Mister Moses, Mister Paul Yiu, Mr Garcia Capitan and all Member

- X_{3},X_{5},X_{13},X_{14} are circumcenter, Nine point center,

the first Fermat point, the secon Fermat point.

- X_{R13in5} is reflection of the X_{13} in X_{5}

- X_{R14in5} is refelection of the X_{14} in X_{5}

- (X_{13}),(X_{14}),(X_{R13in5}),(X_{R14in5}) are four circle with radius equal

radius of circumcicle.

Then:

1-(Dao Thanh Oai-Francisco Javier Garcia Capitan)

X_{R14in5}X_{13}=X_{R13in5}X_{14}=Radius of circumcircle

2-(Dao Thanh Oai) Center of the Lester circle X_{1116} is radical center of (X_{13}),(X_{14}), and circumcircle

3-(Dao Thanh Oai) Circle throungh X_{R13in5}, X_{R14in5}, X_3 tangent with Euler line at X_3, center of this circle  is radical center of (X_{R13in5}),(X_{R14in5}), circumcircle

4-(Dao Thanh Oai) X_{R13in5}, X_{R14in5},X2 are collinear

Best regards

Sincerely

Dao Thanh Oai

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#1259

A little remark on (1):

Lines joining (reflection of X(13) in X(5)) to X(14) and (reflection of X(14) in X(5)) to X(13) are parallel to line joining X(5) and X(39).

Best regards,

Francisco Javier.