Isodynamic point

The two isodynamic points belong to the excellent points of a triangle .

Let a triangle ABC be given with the bisectors of its interior and exterior angles. Let U _a be the intersection of the bisector of the line BC, V _a the point of intersection of the corresponding outer angle bisector with BC. The points U _b and V _b (each on CA) and U _c and V _c (each on AB) are defined accordingly . Then the three circles with the diameters | U _a V _a |, | U _b V _b | and | U _c V _c | two points S and S 'in common. S is called the 1st isodynamic point ( Kimberling number ), S 'is the 2nd isodynamic point (Kimberling number ). ${\displaystyle \alpha }$ ${\displaystyle X_{15}}$ ${\displaystyle X_{16}}$

Coordinates

Isodynamic points ( and ) ${\displaystyle X_{15}}$ ${\displaystyle X_{16}}$
Trilinear coordinates	${\displaystyle \sin \left(\alpha \pm {\frac {\pi }{3}}\right)\,:\,\sin \left(\beta \pm {\frac {\pi }{3}}\right)\,:\,\sin \left(\gamma \pm {\frac {\pi }{3}}\right)}$
Barycentric coordinates	${\displaystyle a\sin \left(\alpha \pm {\frac {\pi }{3}}\right)\,:\,b\sin \left(\beta \pm {\frac {\pi }{3}}\right)\,:\,c\sin \left(\gamma \pm {\frac {\pi }{3}}\right)}$

properties

• The two isodynamic points are isogonally conjugated to the two Fermat points .
• The inversion (mirroring of a circle) at the periphery leads one of the two isodynamic points to the other.
• The base triangles of the two isodynamic points are equilateral .
• The isodynamic points lie on the Brocard axis .
• The three circles are circles of Apollonios

literature

• Roger A. Johnson: Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , pp. 222, 294-297 (first published in 1929 by the Houghton Mifflin Company (Boston) under the title Modern Geometry ).

Web links

Commons : Isodynamic points  - collection of images, videos and audio files

• Eric W. Weisstein ( MathWorld ): Isodynamic Points ( First isodynamic Point and Second isodynamic Point )