Excenters and incenter of a triangle

Geometry Level pending

A ( − 1 , 3 ) , B ( 1 , 5 ) {\rm{A}}\left( { - {\rm{1}},{\rm{ 3}}} \right),{\rm{ B}}\left( {{\rm{1}},{\rm{ 5}}} \right) are two excenters of a triangle PQR and the remaining excenter C lies on the circle x 2 + y 2 − 6 x − 2 y − 10 = 0 {x^2} + {y^2} - 6x - 2y - 10 = 0 then the locus of the incenter of triangle PQR is a circle. Find its radius

20 \sqrt {20} 18 \sqrt {18} 27 \sqrt {27} 24 \sqrt {24}

Vidyamanohar Sharma by Vidyamanohar Sharma , 58, India

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1 solution

Given circle is circum circle of excentral triangle ABC of triangle PQR. Orthocenter of the excentral triangle is the incenter of the triangle. Image of orthocenter in any side lies on the circumcircle. Hence the required locus is x 2 + y 2 + 6 x − 14 y + 38 = 0 {x^2} + {y^2} + 6x - 14y + 38 = 0

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Can you explain how you arrived at the final equation?

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