Please refer to Technology Lab 5-3 for instructions on this construction.
1. Construct a triangle.
2. Construct the perpendicular bisector of each side of the triangle. Construct the point of intersection of these three lines. This is the circumcenter of the triangle. Label it U and hide the perpendicular bisectors.
3. In the same triangle, construct the bisector of each angle. Construct the point of intersection of these three lines. This is the incenter of the triangle. Label it I and hide the angle bisectors.
4. In the same triangle, construct the midpoint of each side. Then construct the three medians. Construct the point of intersection of these three lines. Label the centroid C and hide the medians.
5. In the same triangle, construct the altitude to each side. Construct the point of intersection of these three lines. Label the orthocenter O and hide the altitudes.
6. Move a vertex of the triangle and observe the positions of the four points of concurrency. In 1765, Swiss mathematician Leonhard Euler showed that three of these points are always collinear. The line containing them is called the Euler line.

1. Which three points of concurrency lie on the Euler line?
2. Make a Conjecture Which point on the Euler line is always between the other two? Measure the distances between the points. Make a conjecture about the relationship of the distances between these three points.
3. Make a Conjecture Move a vertex of the triangle until all four points of concurrency are collinear. In what type of triangle are all four points of concurrency on the Euler line?
4. Make a Conjecture Find a triangle in which all four points of concurrency coincide. What type of triangle has this special property?