Let I be the incentre of triangle ABC. Let the lines
AI, BI and CI produced intersect the circumcircle of triangle
ABC at D, E and F respectively. Prove that EF is perpendicular
to AD.
Let \frakC be a circle with centre O and radius k.
For each point P ¹ O, we define a mapping P ®P¢ where P¢ is that point on OP produced for which
|OP ||OP¢| = k^{2} .
In particular, each point on \frakC remains fixed, and the mapping
at other points has period 2. This mapping is called inversion
in the circle \frakC with centre O, and takes the union of the
sets of circles and lines in the plane to itself. (You might want to
see why this is so. Analytic geometry is one route.)
(a) Suppose that A and B are two points in the plane
for which |AB | = d, |OA | = r and
|OB | = s, and let their respective images under the
inversion be A¢ and B¢. Prove that
|A¢B¢| =
k^{2} drs
.
(b) Using (a), or otherwise, show that there exists a sequence
{ X_{n} } of distinct points in the plane with no three collinear
for which all distances between pairs of them are rational.