Distance Ratios
The isodynamic points were originally defined from certain equalities of ratios (or equivalently of products) of distances between pairs of points. If and are the isodynamic points of a triangle, then the three products of distances are equal. The analogous equalities also hold for . Equivalently to the product formula, the distances, and are inversely proportional to the corresponding triangle side lengths, and .
and are the common intersection points of the three circles of Apollonius associated with triangle of a triangle, the three circles that pass through one vertex of the triangle and maintain a constant ratio of distances to the other two vertices. Hence, line is the common radical axis for each of the three pairs of circles of Apollonius. The perpendicular bisector of line segment is the Lemoine line, which contains the three centers of the circles of Apollonius.
Read more about this topic: Isodynamic Point
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