# Irrational Nine-Point Centre is Impossible for a Triangle with Rational Vertices

Chattopadhyay, Siddhartha Sankar (2023) Irrational Nine-Point Centre is Impossible for a Triangle with Rational Vertices. At Right Angles. pp. 59-60. ISSN 2582-1873

## 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 xy-plane 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 nine-point circle, often referred to as the ‘nine- point centre’, for a triangle with rational vertices. The precise statement is as follows.

Item Type: Articles in APF Magazines