Symmedian

Symmedian line II

The following properties of the symmedian make a natural continuation of the subject started in Symmedian_0.html . In particular they characterize point E in the figure below, which is the focus of an Artzt parabola of the triangle (see Artzt.html ).

[1] From C draw parallel CD to the symmedian AJ and line BD intersecting the symmedian at E. Triangles EBA' and AED are similar to ABC. In fact, EBA' is similar to EAD. Later has angle(ADE)=angle(ABC) and angle(EAD)=angle(BAC), because ADCA' is equilateral trapezium and angle(EAD)=angle(EA'C)=angle(ABC).
[2] Previous property is valid for every cevian AA' and its parallel CD and the resulting construction. Nowhere in the proof was needed the symmedian property of AA'. But now it comes. Triangles EBA' and EA'C are also similar. In fact they have both angle(EBA')=angle(EA'C) and corresponding sides A'B/AC = A'A₁/A'A₂ = c/b, A₁, A₂ being the projections of A' on the sides and using the characteristic property of the points on the symmedian (see [3] in Symmedian_0.html ). By [2] we have also EB/EA'=c/b. This proves the similarity of EBA and EA'C together with the following additional facts.
[3] Triangle DEC is isosceles and triangles ADE and A'CE are equal.
[4] E is the middle of AA' and AE² = BE*ED = BE*EC.
[5] Angle(BEC) = 2A and AE bisects BEC.
[6] Triangles BEA and AEC are also similar. In particular angle(ABE)=angle(EAC) and rotating triangle ABE about E so that B moves on AB and the triangle remains all the time similar to BEA, makes vertex A move then on line AC (see Similarly_Rotating.html ).

Symmedian line II

See Also