A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

A SEQUEL TO.EUCLID.

202

A'BC', A'B'C, pass through the point of contact of the
nine-points circle of ABC with its inscribed circle.
Dem.-Let O be the circumcentre. Join OA, and
draw A"E parallel to OA, meeting OlI in E; then EA'
is a radius of the nine-points circle. Let AD be the
bisector of the angle BAC; then the incentres of the

triangles ABC, ABC' are in the line AD. Let these
be I, I'. Join I'A". It is required to prove that I'A"
passes through the point of contact of the nine-points
circle with the incircle of ABC. From I let fall the
perpendicular IL on AB. Join LI'. It is easy to see
that the triangle ILI' is isosceles;-in fact IL is equal
to LI'. Hence if r be the inradius of the triangle ABC,