The triangles inscribed in a fixed circle (c) and having a fixed orthocenter H depend on a real parameter. They build a

That the Euler circle is common to all triangles with fixed H follows trivially from the properties of this circle, having its center E on the middle of HO (O the center of c) and its radius equal to half the radius of c (see Euler.html ).

Take a point A' varying on the circle c and representing the other intersection point with c of the altitude from A. By the properties of the orthocenter side BC is on the medial line of HA' and A is on the extension of HA'. Thus the triangle's position is completely defined from the position of A'. It is also easy to see that {BH,CH} are respectively orthogonal to {AC,BA}. Besides the properties of triangle OHA' show that circle c is a principal (or director) circle of an ellipse with foci at {H,O}, and major axis equal to the radius of c.

If H is on the circle, then the triangle is a right-angled one. The vertical sides pass all the time through H and the hypotenuse passes all the time through O. The enveloping conic is a degenerate one represented by the two bundles of lines through the two fixed points {H, O}.

EulerRelation.html

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