BY HYEONMI LEE

4. Centers of a Triangle

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.

By using Geometer's Sketchpad (GSP)

let's Construct the centroid and explore its location for various shapes of triangles.

One more thing, find the property, which is that the centroid divides each median into two segments so that its ratio is 2:1 from vertex regadless of the kind of 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. (Note: the foot of the perpendicular may be on the extension of the side of the triangle.) It should be clear that H does not have to be on the segments that are the altitudes. Rather, H lies on the lines extended along the altitudes.

Use GSP to construct an orthocenter H and explore its location for various shapes of triangles. Click the above "orthocenter" in order to make sure that the orthocenter for obtuse triangle is out side of the 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, C is on the perpendicular bisector of each side of the triangle. Note: C may be outside of the triangle.

Construct the circumcenter C and explore its location for various shapes of triangles. It is the center of the CIRCUMCIRCLE (the circumscribed circle) of the triangle.

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 I must be on the angle bisector of each angle of the triangle.

Use GSP to find a construction of the incenter I and explore its locationfor various shapes of triangles. The incenter is the center of the INCIRCLE (the inscribed circle) of the triangle.

Now, Let's see how different are the position of centers according to different shaped triangles!!

Equilateral Triangle

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Isosceles Triangle

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