Secant line point a "tangent line" and in no points an "exterior line". A "chord" of a circle is the line segment that joins two distinct points of the circle. A chord is therefore contained in a unique secant line and each secant line determines a unique chord. In rigorous modern treatments of plane geometry, results that seem obvious and were assumed (without statement) by Euclid in his treatment, are usually proved. For instance, "Theorem (Elementary Circular Continuity)": If formula_1 is a circle and formula_2 a line that contains a point that is inside formula_1 and a point that is outside of
Concyclic points the sides of a triangle, then the six points of intersection of the lines and the sides of the triangle are concyclic, in what is called the Lemoine circle. The van Lamoen circle associated with any given triangle formula_6 contains the circumcenters of the six triangles that are defined inside formula_6 by its three medians. A triangle's circumcenter, its Lemoine point, and its first two Brocard points are concyclic, with the segment from the circumcenter to the Lemoine point being a diameter. A polygon is defined to be cyclic if its vertices are all concyclic. For example, all the vertices