[alogo] Symmedian and Vecten

The Vecten configuration of triangle DEF results by constructing squares on the sides of the triangle, called [flanks]. Then extending the sides of the flanks opposite to the triangle sides we get a triangle ABC, similar to DEF. AF is symmedian line of ABC, since the distances of point F from the sides AB, AC are at ratio FM/FH = FD/FE = AB/AC, which is characteristic for the symmedian. Further, AF is also symmedian of DEF through F, since by similarity the symmedian of DEF should divide the angle at F as AF divides the angle at A. Thus the two triangles share the same symmedians, and consequently the same symmedian point K, which is the center of similarity of the two triangles.

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The figure shows also how to solve the exercise of construction of triangle DEF, inversely from the triangle ABC: Take the symmedian point K of ABC and consider the construction of squares inscribed in the three triangles AKB, BKC, CKA.Look at file Symmedian_0.html for the elementary facts about symmedians.

See Also



Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington DC, Math. Assoc. Ammer., 1995.

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