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November 30th, 2012, 10:27 AM  #1 
Senior Member Joined: Aug 2012 From: New Delhi, India Posts: 157 Thanks: 0  Centers of a triangle with rational points Q: Consider a triangle which PQR, where all three of P,Q,R are rational points*. Then which of the following points of the triangle PQR is (are) always rational points*? A) Centroid B) Incentre C) Circumcentre D) Orthocentre * A rational point is a point for which both the x and y coordinates are rational numbers. More than one option may be correct. Source : IITJEE 1998 My attempt at an answer: It is obvious that the centroid G will be a rational point. For the orthocentre H, we know that the slope of lines AB and AC will be rational. Hence slopes of lines perpendicular to them will be rational. Therefore, the altitudes will have rational slope and pass via rational points, hence, their intersection will be a rational point. (This is where I'm confused at, is this statement correct? Please provide a counter example if you can) Let O be the circumcentre , therefore, O is also rational. The incentre's coordinates are dependent on the length of the sides of the triangle, and hence it may not always be rational. Thanks for your help. Regards, Rejjy 01Dec2012 00:57 IST 
December 1st, 2012, 02:40 AM  #2 
Senior Member Joined: Aug 2012 From: New Delhi, India Posts: 157 Thanks: 0  Re: Centers of a triangle with rational points
Bump.......!

December 1st, 2012, 02:41 AM  #3 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs  Re: Centers of a triangle with rational points
I wouldn't bump unless the topic fall off the first page. 

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