## X(15) (1ST ISODYNAMIC POINT)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon , select the center name from the list and, then, click on the vertices A, B and C successively.

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This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           sin(A + π/3) : sin(B + π/3) : sin(C + π/3)
= cos(A - π/6) : cos(B - π/6) : cos(C - π/6)

Barycentrics    a sin(A + π/3) : b sin(B + π/3) : c sin(C + π/3)

Let U and V be the points on sideline BC met by the interior and exterior bisectors of angle A. The circle having diameter UV is the A-Apollonian circle. The B- and C- Apollonian circles are similarly constructed. Each circle passes through one vertex and both isodynamic points. (This configuration, based on the cevians of X(1), generalizes to arbitrary cevians; see TCCT, p. 98, problem 8.)

The pedal triangle of X(15) is equilateral.

X(15) lies on these lines:
1,1251    2,14    3,6    4,17    13,30    18,140    35,1250    36,202    55,203    298,533    303,316    395,549    397,550    532,616    628,636

X(15) is the {X(3),X(6)}-harmonic conjugate of X(16).

X(15) = reflection of X(I) in X(J) for these (I,J): (13,396), (16,187), (298,618), (316,624), (621,623)
X(15) = isogonal conjugate of X(13)
X(15) = isotomic conjugate of X(300)
X(15) = inverse-in-circumcircle of X(16)
X(15) = inverse-in-Brocard-circle of X(16)
X(15) = complement of X(621)
X(15) = anticomplement of X(623)
X(15) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,202), (13,62), (74,16)
X(15) = crosspoint of X(I) and X(J) for these (I,J): (13,18), (298,470)
X(15) = crosssum of X(I) and X(J) for these (I,J): (15,62), (532,619)
X(15) = crossdifference of any two points on line X(395)X(523)
X(15) = X(6)-Hirst inverse of X(16)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.