Assignment #4

 

Centers of Triangles

 

By

 

Michelle Nichols

 

 

 

Centroid

The Centroid of a triangle is the intersection of the medians, or the center of gravity.It is formed by finding the midpoints of the triangle, connecting each midpoint to the vertices, and connecting their intersections.

 

Incenter

The 3 angle bisectors of a triangle intersect at a point called the Incenter of a triangle.It is the center of the largest circle that will fit inside the triangle.Look at the Incenter:

 

Circumcenter

The circumcenter is the intersection of all 3 perpendicular bisectors of the sides of a triangle.Itís called the circumcenter because it is the center of a circle that passes through all of the vertices of the triangle.

 

Orthocenter

The Orthocenter is the intersection of the 3 altitudes of a triangle.An altitude of a triangle is a segment that is drawn so that is passes through the vertex and is perpendicular to the opposite side.

 

Euler Line

The centroid, circumcenter, and orthocenter of a triangle are all collinear.The Euler Line is the line that connects all of these points.

 

Exploration

 

Take any triangle.Construct a triangle connecting the 3 midpoints of the sides.This is called the MEDIAL triangle.It is similar to the original triangle and ľ of its area.Construct G, H, C, and I for this new triangle. Compare to G, H, C, and I in the original triangle.

 

 

Now if we construct the Euler Line for both, thus observing G, H, and C, letís see how they compare.

 

Large Triangle†††† ††††††††††††††††††††††††††††††††††††† †††††††††††††††††† Medial Triangle

†††††††††††††††††††††††††† ††††††††† ††††††††††††††††

The points lie on the same line, however, the locations and distances between each change.The larger trianglesí Euler Line appears to be twice the distance.When measurements and ratios are taken, we find this to be exactly true.The Larger triangle Euler Line has a 2:1 ratio to the medial trianglesí Euler line.Also, the centroid and orthocenter swap sides.

Observe the triangles together:†††††††††††††††††††††††

 

 

 

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