[alogo] Rotating triangle's sides about their middle

Rotate the sides of triangle t = (ABC) about their middles, by an angle (phi). Extending them and taking intersections, defines a new triangle s = (GHI), similar to t. As the angle (phi) varies, the triangle s takes various positions and the following facts are true:
(1) The vertices G, H and I of s move respectively on three fixed circles c1, c2 and c3, which are tangent to the circumcircle c of t, respectively at A, B, C and pass through the center O of c.
(2) The circumcircle d of s is concentric to c.
(3) Segments AG, BH, CI are equal in length and tangent to d.
(4) The vertices L, M and N, of the 2nd Brocard triangle u of t, coincide with the second intersection points of the Brocard circle e with the three circles c1, c2 and c3 of (1).
(5) The symmedians of s pass, at all times, through the vertices of the 2nd Brocard triangle u of t.
(6) The symmedian point P of the triangle s, describes the Brocard circle e of triangle t.


[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]


The proofs rely on the following remarks and the well known proposition, for the variation of similar figures, discussed in the book of Yaglom.
(0) If the figure F moves in such a way that all positions are similar to the original position and so that three lines l, m and n of F, not passing through a common point, pass at all times through three given points, then every line of F passes at all times through some constant point, and every point of F describes a circle.
[I.M.Yaglom, Geometric Transformations II, Random House 1968, p. 72]

(1) Quadrangle EFCI is cyclic and its circumcircle c3 is tangent to the circumcircle c of ABC and passes trhough its center O. Analogous statements hold for the quadrangles DBHE and DGAF. c1, c2, c3 are the circles, referred in (0), described by point G, H, I, as s changes for varying (phi). Using the equality of the radii of these circles and the equality (to phi) of the angles at D, E and F we see the equality of segments CI, BH and AG. A consequence is also that OI, OH and OG are equal and orthogonal to the previous three segments respectively. This shows that the circumcircle d of GHI is concentric to that of ABC and that the CI, BH, AG are tangent to d. This proves (1), (2) and (3).
(2) The symmedian CK cuts the Brocard circle at M and OM is orthogonal to CK (M viewing the diameter OK under a right angle). Thus M is the middle of the symmedian viewed as a chord of the circumcircle c. This shows that R is the other than O intersection point of c with c3. The angle EIM being constant (for varying phi) the symmedian PI, of all triangles s passes through M. Analogous statement holds for the other symmedians PN, PL of the varying triangle s. On the other hand P, viewing MN under a constant angle moves on the Brocard circle. The Brocard circle is the circle, referred in (0), described by the symmedian point P of s, vor varying (phi).
Look at RotatingSides2.html , for a continuation of the subject.


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