You've already learned about four points of concurrency: circumcenter, incenter, orthocenter, and centroid. On this web page you will discover how these points relate to a special line, the Euler line. The Euler line is named after the Swiss mathematician Leonhard Euler (1707–1783), who proved the relationship that you will discover.

Sketch

This sketch shows a triangle and its four points of concurrency. You can change the shape or size of the triangle by dragging its vertices. If you drag a point off the screen, press Start Over to return it.

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Investigate

1. Change the shape and size of the triangle and look for a relationship among the points of concurrency. Try scalene, isosceles, and equilateral triangles. What do you notice?
   
   Press one of the buttons to show the lines connecting two of the triangle's centers. Press the button again to hide the line. Hide Lines will hide all of the lines.
2. What do you notice? Which three points of concurrency are collinear?
3. Formulate the Euler Line Conjecture: The _____, the _____, and the _____ are the three points of concurrency that always lie on a line.
4. Are the three collinear points always in the same order? If so, which point is always between the other two?
5. Under what circumstances are all four points collinear?

Sketch

The three special points that lie on the Euler line determine the Euler segment. The point of concurrency between the two endpoints of the Euler segment divides the segment into two smaller segments. This sketch shows the Euler segment, the lengths of the two parts, and the ratio of the lengths. You can change the shape or size of the triangle by dragging its vertices. If you drag a point off the screen, press Start Over to return it.

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Investigate

1. Which points of concurrency are represented by points A, B, and C? Use your knowledge of points of concurrency along with what you learned in the first sketch on this page.
2. Is one part of the Euler segment always the shorter part? If so, which part?
3. What do you notice about the ratio AB/BC?
4. Formulate the Euler Segment Conjecture: The _____ divides the Euler segment into two parts so that the smaller part is _____ the larger part.