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The Euler line theorem and the nine-point-circle theorem are two extraordinary theorems that are generally overlooked in high school geometry classes. The former theorem was not proved until 1765; and the latter, not until 1820. Yet geometry students are well acquainted with three concurrence theorems that set the stage for the two forementioned theorems: In any trianglethe perpendicular bisectors of the sides intersect at a point, the circumcenter, which is the center of the circumscribed circle;
the medians intersect at a point, the centroid, which is two-thirds the distance from any vertex to the midpoint of the opposite side; and the lines containing the altitudes intersect at a point called the orthocenter. These facts can readily be discovered by experimental constructions, and they are among the more interesting results that can be proved in geometry classes.
The Euler line theorem pulls these concurrencies together by establishing that the circumcenter, the centroid, and the orthocenter are collinear; the ninepoint-circle theorem extends this sequence of theorems to the remarkable conclusion that nine other well-defined points lie on a circle centered at the midpoint of the segment joining the circumcenter and the orthocenter. The proofs of these two theorems by coordinate geometry or by the traditional methods of high school geometry are laborious. But by turning to a geometric function, we can offer concise, accessible, and elegant proofs.
Functions are fundamental building blocks for all of mathematics, but unfortunately, functions play an insignificant role in most high school geometry courses. Modern geometry textbooks often include a chapter on geometric functions, or transformations; but it is often the last chapter in the book, so few teachers reach it by the end of the year. Those who do get there find an introduction that merely describes several types of transformations and calls on students to construct images of some geometric figures. Applications are seldom mentioned, and the students' reaction, with some justification, is often, "So what?" In this article we present a train of ideas leading to the application of a particular transformation, a dilation, to the proofs of both the Euler line and nine-point-circle theorems. In so doing, we hope to help introduce students to the important role that functions can play in the field of geometry.
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