The Euler line of a triangle is a line going through several important triangle centers, including the orthocenter, circumcenter, centroid, and center of the nine point circle. The fact that such a line exists for all non-equilateral triangles is quite unexpected, made more impressive by the fact that the relative distances between the triangle centers remain constant.
The simplest proof makes use of homothety, specifically the one with center at the centroid of the triangle, with scale factor . This sends vertices to the midpoints of the opposite sides, since the centroid divides the medians in a 2:1 ratio, meaning that triangle is sent to the medial triangle.
Therefore, the orthocenter of triangle is sent to the orthocenter of the medial triangle, which is the circumcenter of . As a result, the orthocenter, circumcenter, and centroid are all collinear, as desired.
The proof above shows more than collinearity: since is sent to through a scale factor of , we have . In fact, more is true: if is the nine-point center of the triangle, we have
The Euler line also contains dozens of other triangle centers, some of which have only been discovered recently. Of special note are the following points:
- The De Longchamps point the reflection of the orthocenter over the circumcenter.
- The Exeter point defined as follows: if is the tangential triangle of , and are the intersections of the medians of with the circumcircle of , then concur at the Exeter point.
- The Schiffler point: if is the incenter of , then the Euler lines of , and concur at the Schiffler point.
Additionally, if is the foot of the altitude from to and is the midpoint of (with and defined analogously), then
The intersection of and lies on the Euler line.
The slope of the Euler line relates to the slope of the sides in a nice way: If are the slopes of the three sides of a triangle , and is the slope of the Euler line, then
One important consequence of the Euler line is that information about any one of the centroid, orthocenter, and circumcenter can be derived from information about the other two. For example,
Points and are randomly chosen on a unit circle. The locus of the orthocenter of is a region . What is the area of ?
Instead of focusing on the orthocenter, it helps to focus on the other two major triangle centers: the centroid and the circumcenter. The circumcenter is always the center of the unit circle, so it is only necessary to note that the centroid can lie anywhere within the unit circle, and nowhere else (why?). Since , this implies that the maximum possible value of is 3, and the region is a circle of radius 3. The area of is thus .
Another result due to the properties of the Euler line:
If is a triangle with circumcenter and orthocenter , then one of the areas of , , and is the sum of the areas of the other two.
Mohit, influenced by Akshat and Anshuman, took a right (right angled at ), where and , and started doing aimless constructions, the steps of which are given below:
() He drew perpendicular bisector of which intersects at a point and another perpendicular bisector of which intersects at a point .
() Then he constructed a circle taking center and radius of the circle as . Then he constructed an which is equal to such that lies on the circle.
() He then joined which meets at .
() Then he joined which intersects at and at .
Mohit then wondered what could possibly be equal to.