Let A, B, and P be three fixed, non-collinear points. Let C be a point on ray AP different from A. The incircle of ΔABC touches BC at D and AC at E.

Prove that the line DE passes through a fixed point as C varies on ray AP.

Let |BD| = b and |CD| = |CE| = c.
Let F be a fixed point on AC such that |AF| = |AB|, then |EF| = b.
Let M be the mid-point of BF.
Let d and e be unit vectors in the directions of CD and CE.

   Vector  EM      = 0.5 (EF + EB)
                        = 0.5 (EF + EC + CD + DB)
                        = 0.5 [(|EF| + |EC|)(-e) + (|CD| + |DB|)d]
                        = 0.5 [(b + c)(-e) + (c + b)d]
                        = 0.5 (b + c)(d - e)
                        = 0.5 (b/c + 1)(cd - ce)
                        = 0.5 (b/c + 1)(CD - CE)
                        = 0.5 (b/c + 1) ED

Thus EM is parallel to ED, and the fixed point M must always lie on ED, whatever the values of b and c; i.e. for all positions of C on the ray AP.

Posted by Harry on 2010-12-11 00:53:18