. Here is the expression that appears in the relaxed Newton’s method with relaxation parameter . A similar construction exists for rational functions instead of polynomials. Construction The circle of Apollonius through vertex of triangle may be constructed by finding the two (interior and exterior) angle bisectors of the two angles formed by lines and at vertex , and intersecting these bisector lines with line . The line segment between these two intersection points is the diameter of the circle of Apollonius. The isodynamic points may be found by constructing two of these circles and finding their two intersection points. Another compass and straight-edge construction involves finding the reflection of vertex across line (the intersection of circles centered at and through ), and constructing an equilateral triangle inwards on side of the triangle (the apex of this triangle is the intersection of two circles having as their radius). The line crosses the similarly constructed lines and at the first isodynamic point. The second isodynamic point may be constructed similarly but with the equilateral triangles erected outwards rather than inwards. Alternatively, the position of the first isodynamic point may be calculated from its trilinear coordinates, which are The second isodynamic point uses trilinear coordinates with a similar formula involving in place of . Notes References . . . . . . . . . . . The definition of isodynamic points is in a footnote on page 204. . The discussion of isodynamic points is on pp. 138–139. Rigby calls them "Napoleon points", but that name more commonly refers to a different triangle center, the point of concurrence between the lines connecting the vertices of Napoleon's equilateral triangle with the opposite vertices of the given triangle. . See especially p. 498. External links Isodynamic points X(15) and X(16) in the Encyclopedia of Triangle Centers, by Clark Kimberling Triangle centers