the radius. In like manner, if OC be bisected in K, then IK =
the radius. Hence we have the following theorem:—The nine points made up of the feet of the perpendiculars, the middle points of the sides, and the middle points of the lines from the vertices to the orthocentre, are concyclic.
Def.—The circle through these nine points is called the “nine points circle” of the triangle.
5. The circumcircle of a triangle is the “nine points circle” of each of the four triangles formed by joining the centres of the inscribed and escribed circles.
6. The distances between the vertices of a triangle and its orthocentre are respectively the doubles of the perpendiculars from the circumcentre on the sides.
7. The radius of the “nine points circle” of a triangle is equal to half its circumradius.
PROP. VI.—Problem.
In a given circle (ABCD) to inscribe a square.
