Proof by Antreas Varverakis.

-Consider the circumcenters K, J of the triangles BDF and BEG, respectively. Let DM, EL be diameters of these circles, respectively. The following remarks conclude to a proof.

- Orthogonal triangles EGL, DFM are similar, the angles at M, L being equal to that at B.

- As E, D vary on lines BC, BA respectively, points M, L move on lines BM, BL, orthogonal to BA, BC respectively.

- Triangles EFN, GND are similar.

- Triangles FND, ENG are similar.

- Triangle RST is similar to the above FND, ENG.

- Angles FND, ENG and LBM are equal (pi-beta).

- The angles SRT = NEG = NBG and STR = NDF = NBE.

- Angle RTG = UTG + RTU = BFG+NBG.

- RT orthogonal to FG <= > RTU = NBG = pi/2 - f <= > BN passes through the circumcenter of BFG.

- E, D project on ML to the intersection-points of ML with the circles.

When the middle R of ED is on the medial line of the triangle's basis FG, then the line ED is tangent to an interesting parabola, studied in MedialParabola.html .

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