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Locus problem (Posted on 2007-05-27) Difficulty: 3 of 5
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?

See The Solution Submitted by Bractals    
Rating: 4.0000 (1 votes)

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Unproven solution Comment 1 of 1
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

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