Concurrent circles and concurrent hyperbolas
- Dear Hyacinthos,
Let`s consider the triangle ABC and points Y, Z on the sidelines
CA, AB. Let P be the point so that the angles <AYP and <AZP are
1.If the angles <AYP and <AZP have the same orientation then the
geometric locus of P is the circumcircle Ca of the triangle AYZ
2.If the angles <AYP and <AZP have different orientation then the
geometric locus of P is a hyperbola Ha going through the points
A, Y, Z.
Now let's consider the triangle ABC, points X, Y, Z on the
sidelines BC, CA, AB.
1.If we have angles with the same orientation let Ca, Cb, Cc the
circumcircles of the triangles AYZ, BZX, CXY. The circles Ca,
Cb, Cc are concurrent in P. If P is on the circumcircle of the
triangle ABC then X, Y, Z are collinear points.
2.If we have angles with different orientation let Ha(going through
A, Y, Z), Hb(going through B, Z, X), Hc(going through C, X, Y) be
the hyperbolas. The hyperbolas Ha, Hb and Hc are concurrent in
point P if and only if XYZ is the pedal triangle of the point P.
If P is on the circumcircle of the triangle ABC then X, Y, Z are
collinear points on the Simson's line of P (S(P)).