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November 30th, 2012, 10:27 AM   #1
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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 co-ordinates are rational numbers.
More than one option may be correct.
Source : IIT-JEE 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 co-ordinates 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
01-Dec-2012
00:57 IST
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December 1st, 2012, 02:40 AM   #2
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Re: Centers of a triangle with rational points

Bump.......!
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December 1st, 2012, 02:41 AM   #3
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Re: Centers of a triangle with rational points

I wouldn't bump unless the topic fall off the first page.
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