[alogo] Reflecting a Steiner line

Consider a line XH passing through the orthocenter H of triangle ABC. Reflect this line on the sides of ABC and show that the three reflected lines concur at a point Y on the circumcircle of the triangle.

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The reflected on AC line EY and the reflected on AB line FD pass correspondingly from the reflected of H on the respective sides, points E, F on the circumcircle (see Orthocenter.html ).
The angle of these lines at Y is equal to the corresponding angle(FDE) of the triangle DEF, formed by the reflections of H on the sides of ABC. This follows by measuring the angles:

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and using the well known properties of DEF (ibid). Letting the variable line HX pass not through H but through an arbitrary fixed point H', yields another interesting configuration studied in OrthicAndPedal.html .

Remark Note that point Y is the one on the circumcircle for which the corresponding Steiner line coincides with HX. The proof could be given by showing that the reflected points of Y on the sides of ABC are on line HX. This is discussed in SteinerLine.html .

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