This is a STILL image.

It is NOT a movie.


As in the previous task, we can think of B and C as fixed, along with the angle at B.
Again, as point A moves to the right, the angle at A gets smaller so the slope of the bisector of angle A gets closer to the horizontal, ie the line 'dips down'. This might suggest that as A moves to the right, P moves downwards. However, it turns out that this is not the case.

There is a theorem which states that if AP is the bisector of angle A, then AB/AC = BP/CP (see GEOdd).
What happens to AB/AC as A moves further and further to the right? What does this say about P ?

A related approach is to extend CA to a point D such that AD = AB. It then turns out that BD is parallel to PA (this is fairly easy to derive once one has drawn a diagram!). In turn this means that triangle CDB is an enlargement of triangle CAP with scale factor CD/CA = CB/CP. What happens to this scale factor as A moves further and further to the right, ie as BA gets larger and larger, and hence CA and CD get larger and larger?

A simpler, more qualitative approach is to consider extremes, namely the case where A is very close to B, so that P is 'as close as you like' to B ('beliebig' close), and the case where A is very far from B so that triangle ABC is almost isosceles and so that P is near the midpoint of BC.

We show another related task on the next and final page: PS-3

a high-res pdf file of this new GEOaa-zz task