## Assignment 4 Centers of a Triangle

#### by Kelli Nipper

The CENTROID (G) of a triangle is the common intersection of the three medians. A median of
a triangle is the segment from a vertex to the midpoint of the opposite side.
• The centroid divides each median into two parts, the ratio of whose lenghts is 2/1.
• The six small triangles formed by the medians have equal area.

#### Click Here to view the centroid's location for various shapes of trianlges.

MEDIAL TRIANGLE
A midsegment that connects the midpoints of two sides is parallel to the third side and half its length. By connecting the midpoints of the sides of any triangle, you create a similar triangle that has one-fourth of the original triangles area and the location of the centroid for both triangles is the same. If you continue to make successive medial triangles, these "in-triangles" converge to the centroid of the triangle.

The ORTHOCENTER(H) of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side.

#### Click Here to view the orthocenter's location for various shapes of triangles.

ORTHIC TRIANGLE
By connecting the feet of the altitudes of any acute triangle, you create a similar triangle that has the same position fo the othocenter as the original triangle.

The CIRCUMCENTER(C) of a triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies on the perpendicular bisector of the segment determined by the two points, the orthocenter is on the perpendicular bisector of each side of the triangle.
• The three pairs of angles formed by the angle bisectors are equal to the angles in the triangle.

#### Click Here to view the circumcenter's location for various shapes of triangles.

CIRCUMSCRIBING A TRIANGLE
The circumcenter is equidistant the three vertices; therefore, it is the center of a circle that goes through these points. Using the circumcenter of the triangle as a center and a vertices as a point, a circumcircle can be created.
• The hypotenuse of a right triangle is a diameter of the circumscribing circle.

The INCENTER(I) of a triangle is the point on the interior of the triangle that is equidistant from the three sides. Since a point interior to an angle that is equidistant from the two sides of the angle lies on the angle bisector, then the incenter must be on the angle bisector of each angle of the triangle.

#### Click Here to view the incenter's location for various shapes of triangles.

INSCRIBING A TRIANGLE
The distance from the incenter to each side(along the angle bisectors) is the same. Using the incenter of the triangle as a center and the midpoint of a side as a point, a circle can be created. However, if the triangle is changed the circle doesn't stay inscribed. Using a line through the incenter, perpendicular to any side, an incircle constructed through this incenter and the intersection will stay inside of any triangle.