Radical Axis - Determinant Calculation

Determinant Calculation

If the circles are represented in trilinear coordinates in the usual way, then their radical center is conveniently given as a certain determinant. Specifically, let X = x : y : z denote a variable point in the plane of a triangle ABC with sidelengths a = |BC|, b = |CA|, c = |AB|, and represent the circles as follows:

(dx + ey + fz)(ax + by + cz) + g(ayz + bzx + cxy) = 0
(hx + iy + jz)(ax + by + cz) + k(ayz + bzx + cxy) = 0
(lx + my + nz)(ax + by + cz) + p(ayz + bzx + cxy) = 0

Then the radical center is the point

\det \begin{bmatrix}g&k&p\\ e&i&m\\f&j&n\end{bmatrix} : \det \begin{bmatrix}g&k&p\\ f&j&n\\d&h&l\end{bmatrix} : \det \begin{bmatrix}g&k&p\\ d&h&l\\e&i&m\end{bmatrix}.

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