Math Forum - Ask Dr. Math
The circumcenter O of A B C is just the orthocenter of the medial triangle A ′ B ′ C ′, which, obviously, shares its centroid G with A B C. Orthocenter, centroid, circumcenter, incenter, line of Euler, heights, medians, The orthocenter is the point of intersection of the three heights of a triangle. The amazing Euler Line which contains the Centroid, Orthocenter & Circumcenter.
Thus, the radius of the circle is the distance between the circumcenter and any of the triangle's three vertices. It is found by finding the midpoint of each leg of the triangle and constructing a line perpendicular to that leg at its midpoint.
Where all three lines intersect is the circumcenter. 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 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.
Relation between orthocentre, circumcentre and centroid?
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.
Check out the cases of the obtuse and right triangles below. In the obtuse triangle, the orthocenter falls outside the triangle.
Orthocenter, Centroid, Circumcenter and Incenter of a Triangle
In a right triangle, the orthocenter falls on a vertex of the triangle. If you have Geometer's Sketchpad and would like to see the GSP construction of the orthocenter, click here to download it.
It is the point forming the origin of a circle inscribed inside the triangle. Like the centroid, the incenter is always inside the triangle. It is constructed by taking the intersection of the angle bisectors of the three vertices of the triangle. The radius of the circle is obtained by dropping a perpendicular from the incenter to any of the triangle legs.
It is pictured below as the red dashed line. A median is a segment constructed from a vertex to the midpoint of the subtending side of the triangle. The orthocenter H of a triangle is the point of intersection of the three altitudes of the triangle.
An altitude is a line constructed from a vertex to the subtending side of the triangle and is perpendicular to that side. It should be noted that the orthocenter, in different cases, may lie outside the triangle; in these cases, the altitudes extend beyond the sides of the triangle.
The circumcenter C of a triangle is the point of intersection of the three perpendicular bisectors of the triangle. A perpendicular bisector is a line constructed at the midpoint of a side of a triangle at a right angle to that side. It should be noted that the circumcenter, in different cases, may lie outside the triangle; in these cases, the perpendicular bisectors extend beyond the sides of the triangle.Incenter, Circumcenter, Orthocenter & Centroid of a Triangle - Geometry
The incenter I of a triangle is the point of intersection of the three angle bisectors of the triangle. An angle bisector is a line whose points are all equidistant from the two sides of the angle. Thus, the incenter I is equidistant from all three sides of the triangle. Let X be the midpoint of EF. Construct the median DX. Since G is the centroid, G is on DX by the definition of centroid.
Also, construct the altitude DM.