Valerie Russell

Assignment 4 - Centers of a Triangle

All of my chidren are bringing in peanut butter cookies!

 Fooled you didn't I? I just introduced you to an easy way to remember the meanings of the 4 points of concurrency in a triangle which are the orthocenter, centroid, incenter, and circumcenter. All of : Altitudes - orthocenter My children: Medians - centroid Are bringing in: Angle bisectors - incenter Peanut butter cookies: perpendicular bisectors - circumcenter

Equilateral Triangles

Did you know that each angle in an equilateral tiangle is the vertex angle of an isosceles triangle? Remember, an isosceles triangle has at least two congruent sides.

Did you also know that when you construct three angle bisectors in an equilaeral triangle you also have the altitudes, perpendicular bisectors, and medians of a triangle?

 The 3 angle bisectors divide the equilateral triangle into six overlapping 30-60 right triangles and six smaller non-overlapping 30-60 right triangles. The point of concurrency of the angle bisectors in a triangle is the incenter. Each point on the angle bisector is equidistant from the sides of the angle. The point of concurrency of the perpendicular bisectors in a triangle is the circumcenter. The circumcenter is the center of a circle circumscribed about the triangle. The point of concurrency of the medians in a triangle is the centroid. The centroid of a triangle divides each median into two parts so that the distnce from the centroid to the vertex is twice the distance from the centroid to the midpoint. The point of concurrency of the altitudes in a triangle is the orthocenter.

The Incenter of a Triangle

 The incenter is the center of the inscribed circle and segment mi is the radius of the inscribed circle. It does not matter if the triangle is obtuse, right, or acute, the incenter will always be found on the inside of the triangle.

Orthocenters of different triangles

The orthocenter of a triangle is the point of concurrency of the three altitudes of the triangle. The orthocenter is not always found on the interior of a triangle. In an obtuse triangele the orthocenter is found outside of the triangle. In a right triangle. it is located on the right angle. In an acute triangle, the orthocenter is located on the interior of the triangle.

The Circumcenters of triangles

The circumcenter of a triangle is the point of concurrency of the perpendicular bisectors of the sides of the triangle. Notice the circumcenter appears at different places. In an obutse triangle the circumcenter is located outside of the triangle. In a right triangle it is located on the hypotenuse. In an acute triangle the circumcenter is located inside the triangle.

The Centroid of a triangle

The centroid of a triangle is the point of concurrency of the 3 medians of a triangle. It is always found on the interior of the triangle.