Question
Wed February 01, 2012 By:
 

Let ABC be a triangle in which no angle is 90 . For any point P in the plane of the triangle, let E,F,G denotes reflection of P in the sides BC,CA,AB respectively . Prove the following statements: (a) If P is the incentre or an excentre of ABC , then P is the circumcentre of E,F,G; (b) If P is the circumcentre of ABC , then P is the orthocenter of E,F,G; (c) If P is the orthocenter of ABC , then P is either the incentre or an excentre of E,F,G.

Expert Reply
Sun February 05, 2012
Take the three sides to be AB as x - y + 1 =0 , AC as x + y - 1 =0 and BC as y = 0 ,
Then we can take any arbitrary point P one by one in all the four quadrants (a,b)(-a,-b)(a,-b)(-a,b)
For each point P find the reflections in AB BC CA , to get E F G .
Thus it becomes much simpler to prove for each case by calculating the othocenter , circumcenter and incenter of E F G   and then comparing with P
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