A triangle has two of its sides along the axes. Its 3rd side touches the circle . Find the equation of locus of the circumcentre of the triangle.

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- Apr 30th 2009, 08:32 AMfardeen_genLocus of circumcentre?
A triangle has two of its sides along the axes. Its 3rd side touches the circle . Find the equation of locus of the circumcentre of the triangle.

- May 1st 2009, 12:37 AMearboth
I can't provide you with the equation of the locus but I can give you a few hints:

1.

The circle touches both coordinate axes.

2. You are dealing with a right triangle where the circumcentre is the midpoint of the hypotenuse.

3. If the tangentpoint is T(t, p) then the tangent has the equation:

4. If the tangent is parallel to the coordinate axes then there doesn't exist a circumcentre. Therefore the straight lines x = a and y = a must be asymptotes of the locus.

5. I've drawn a sketch: Triangles in blue, locus in red, asymptotes in green - May 2nd 2009, 09:20 AMpankaj
Let the circumcentre of the triangle Therefore the the three vertices of the triangle will be and The Incentre of the triangle will be obviously as can be seen in the diagram supplied by earboth.

Thus

Recall that the coordinates of the incentre of the triangle having vertices as and is given by

where

On simplification,we get

Therefore the required locus is

This is the equation of the curve which earboth has drawn in red.