Euler segment
Leonhard Euler was one of the most prolific Mathematicians in history.
In this task you will construct an Euler segment in a triangle and investigate some of its features.
Open GeoGebra and then follow the instructions How to construct an Euler segment (.PDF 127KB).

The centroid, or centre of gravity, can never escape the boundary of the triangle. What about the other two points?

Under what conditions will these three special points collide?

Under what conditions will an extension of the Euler segment pass through a vertex of the triangle?

The three points appear to be collinear. Investigate this idea. In what ratio does the centroid appear to cut the segment? This is known as Euler’s conjecture and is difficult to prove.
The incentre of any triangle is the point of intersection of the three angle bisectors of the triangle. 
Design a new GeoGebra file that locates the incentre of a triangle.
Consider: Who needs to ‘know about’ the points you have created?