Chattopadhyay, Siddhartha Sankar
(2023)
Irrational NinePoint Centre is Impossible for a Triangle with Rational Vertices.
At Right Angles.
pp. 5960.
ISSN 25821873
Abstract
In the Iranian Mathematics competition at The University of Isfahan in March 1978, the following problem was given.
Problem 1. [1, 1.6.1] In the xyplane a point is called ‘rational’ if both of its coordinates are rational.
Prove that if the centre of a given circle in the plane is not rational, then there are at most two rational points on the circle.
For the sake of convenience, we call a point ‘irrational’ if it is not rational as per the definition in Problem 1. The idea to solve Problem 1 is to assume, for the sake of contradiction, that there are three rational points on a circle whose centre is irrational and then arrive at a contradiction. Therefore, we can reformulate the above problem and assert that the circumcentre of a triangle is rational if the vertices of the triangle are all rational. In this article, we prove an analogous result for the centre of the ninepoint circle, often referred to as the ‘nine point centre’, for a triangle with rational vertices. The precise statement is as follows.
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