Ideas in Geometry/Instructive examples/Lesson 17- Properties of Triangles
There are different ideas about what the center of a triangle might be. Consider the following. Which do you think best describes the "center" of a triangle? :
1. The perpendicular bisectors of the the 3 sides of the triangle that meet at one point= Circumcenter At this point used as the center, a circle can be made to intersect each of the 3 vertices. This line may not always lie inside of the triangle. This site helps us visualize the circumcenter as well as help construct one: http://www.mathopenref.com/constcircumcenter.html
2. The interior bisectors of each angle meet at a point= Incenter At this point used as the center, a circle can be drawn within the triangle that makes the circle tangent to each of the sides. This circle made is called the incircle. This site helps us visualize the incenter as well as help construct one: http://www.mathopenref.com/constincenter.html
3. The meeting point of the 3 altitudes of a triangle= orthocenter What is an altitude? An altitude is a line from the vertex to the opposite line. This line is perpendicular to the opposite line and a right angle is always formed between the line made and the opposite line of the triangle. The orthocenter may not lie inside of the triangle This site helps us visualize the orthocenter as well as help construct one: http://www.mathopenref.com/constorthocenter.html
4. The meeting point of the lines of a triangle that connect the vertex to the midpoint of the opposite side= Centroid The centroid is unique because it is called the "center of mass." This means the triangle can be balanced on a plane at this point. This site helps us visualize the centroid as well as help construct one: http://www.mathopenref.com/constcentroid.html
But how is this different than an orthocenter? The lines used to create the centroid do not necessarily form a right angle with the other side. When constructing the othocenter, the altitudes always form right angles with the opposite sides.
Helpful acronym to remember which constructions always lie inside the triangle : COIN C ircumenter O rthocenter I ncenter N centroid IN is always inside the triangle!