The circumcenter is not always inside the triangle. In fact, it can be outside the triangle, as in the case of an obtuse triangle, or it can fall at the midpoint of the hypotenuse of a right triangle. See the pictures below for examples of this. You see that even though the circumcenter is outside the triangle in the case of the obtuse triangle, it is still equidistant from all three vertices of the triangle.
If you have Geometer's Sketchpad and would like to see the GSP construction of the circumcenter, click here to download it. The orthocenter is the center of the triangle created from finding the altitudes of each side. The altitude of a triangle is created by dropping a line from each vertex that is perpendicular to the opposite side. An altitude of the triangle is sometimes called the height. Remember, the altitudes of a triangle do not go through the midpoints of the legs unless you have a special triangle, like an equilateral triangle.
Like the circumcenter, the orthocenter does not have to be inside the triangle. For each of those, the "center" is where special lines cross, so it all depends on those lines!
Draw a line called a "median" from each corner to the midpoint of the opposite side. Where all three lines intersect is the centroid , which is also the "center of mass":. Try this: cut a triangle from cardboard, draw the medians. The three perpendicular bisectors of the sides of a triangle meet at the circumcenter.
The circumcenter is also the center of the circle passing through the three vertices, which circumscribes the triangle.
This circle is sometimes called the circumcircle. The orthocenter is the point of intersection of the altitudes of the triangle, that is, the perpendicular lines between each vertex and the opposite side. When the orthocenter is combined with the three vertices, any one of the points is the orthocenter of the other three. To draw the circumcenter create any two perpendicular bisectors to the sides of the triangle. The point of intersection gives the circumcenter.
A bisector can be created using the compass and the straight edge of the ruler. Set the compass to a radius, which is more than half the length of the line segment. Then make two arcs on either side of the segment with an end as the center of the arc. Repeat the process with the other end of the segment. The four arcs create two points of intersection on either side of the segment. Draw a line joining these two points with the aid of the ruler, and that will give the perpendicular bisector of the segment.
To create the circumcircle, draw a circle with the circumcenter as the center and the length between circumcenter and a vertex as the radius of the circle. Incenter: Incenter is the point of intersection of the three angle bisector s.
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