This wiki page shows some simple examples to solve triangle centers using simple properties like circumcenter, Fermat point, Brocard points, incenter, centroid, orthocenter, etc.
One should be able to recall definitions like
- circumcenter the point of which is equidistant from all the vertices of the triangle;
- incenter the point of which is equidistant from the sides of the triangle;
- orthocenter the point at which all the altitudes of the triangle intersect;
- centroid the point of intersection of the medians of the triangle.
Show that if the orthocenter and the incenter of a triangle coincide, then this triangle must be equilateral.
Consider vertex . Let these points coincide at .
Then, we know that is the angle bisector of , and it is also the perpendicular to . Thus, we obtain that .
Since this is true from any vertex, it means that all three angles are equal, and thus we have an equilateral triangle.
Let , , denote the distance between the incenter and the vertices of triangle respectively. Prove that .
Solution: (to be continued)
Prove the Euler line, which states that is collinear.
How many triangles are there with integer side lengths such that the area of the triangle formed by joining the orthocenter, the circumcenter and the centroid of is square units?
Details and assumptions:
The orthocenter of is the point at which the altitudes of intersect.
The circumcenter of is the point which is equidistant from , and .
The centroid of is the point at which the medians of intersect.