## 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.
#### Click Here to explore a medial 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.
#### Click Here to explore an orthic 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.

#### Click Here to explore a circumcircle.

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.
#### Click Here to explore an inscribed triangle.

The **NINE-POINT CIRCLE** for any triangle passes through
the three mid-points of the sides, the three feet of the altitudes, and
the three midpoints of the segments for the respective vertices to the orthocenter.
#### Click Here to explore a Nine-Point Circle

The **EULER LINE** for any triangle is the connection of the centroid,
orthocenter, and circumcenter. The incenter also falls on this line for
isosceles triangles. All four points are coincident in an equilateral triangle.
#### Click Here to explore the Euler Line for
various shapes of triangles.