Let point D lie on side BC of triangle ABC.
Let C₁ and C₂ be the incircles of triangles ABD and ACD respectively.
Let m be the common external tangent to C₁ and C₂ different from BC.
Let P be the intersection of line AD with m.

What is the locus of point P as point D varies between B and C?

I drew this in a CAD program.

My triangle ABC was scalene, BC being the base and longest side.

I constructed the incircle of ABC and the points where AB and AC were tangent to it I designate X and Y respectively.

As D moves towards B C2 increases in size and in the limit P would become X.
As D moves towards C C1 increases in size and in the limit P would become Y.

I constructed AD such that D was at the midpoint of BC.

Now the points X, Y and P lie on some kind of curve; parabola, ellipse, circle, sine/cosine curve?  Without closer examination all are legitimate assumptions.

In my sketch the three points appeared to lie on a circle centred at A.

I am claiming that the locus of P is an arc, centred on A and having a radius that is the distance from A to the tangent points on AB (X) and AC (Y) of the incircle of ABC.

My drawing was sufficiently accurate to confirm to me my assumption. 
I have not done the maths to prove it.
Maybe someone else?

Edited on June 1, 2007, 5:08 am

Posted by brianjn on 2007-06-01 05:04:41