Isodynamic point

[

$S$ and $S^\{prime\}$ as common intersection points of circles of Apollonius.] In Euclidean geometry, every triangle has two isodynamic points, usually denoted as $S$ and $S^\{prime\}$. These points are the common intersection points of the three circles of Apollonius associated with the triangle; hence, the line through these points is the common radical axis for these circles. The centers of these circles are collinear; they all fall on the Lemoine line, which is perpendicular to the radical axis defined by the isodynamic points.

The isodynamic points have other interesting geometric properties, e.g.,

* Inversion with respect to an isodynamic point transforms the original triangle into an equilateral triangle.

* The pedal triangle of an isodynamic point is also equilateral.

ee also

* Circles of Apollonius

References

* Johnson RA (1960) "Advanced Euclidean Geometry", Dover.