Poristic

1. Common orthocenter

A poristic system of triangles is a family of triangles having the same circumcircle and the same incircle. All these triangles depend on one parameter and their discussion starts with the file EulerRelation.html .
The triangles inscribed in a fixed circle (c) and having a fixed orthocenter H depend on a real parameter. They build a poristic family of triangles. The triangles have also a common Euler circle and their sides are tangent to a conic whose shape depends on the position of the orthocenter with respect to the circumcircle.

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.

2. Common orthocenter II

Previous reasoning is valid in the case H is inside the circle.
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}.

3. Common orthocenter III

Orthocenter H lying outside the circumcircle characterizes triangles with an obtuse angle. An analogous argument shows that the sides of ABC are tangent to the hyperbola with focus at {H,O} and major axis the circumradius of the triangle.

Remark The triangle conic in discussion has perspector the triangle center X₂₆₄, which is the isotomic conjugate of the circumcenter.

1. Common orthocenter

2. Common orthocenter II

3. Common orthocenter III

See Also