Assignment 4: Center Secrets of  Triangles

Construct four centers of a triangle, the Centroid (G), the Orthocenter (H), the Circumcenter  (C) and the Incenter (I).

The Centroid is the intersection of the three medians.

The dotted lines are the medians. A median is a line segment that connects the vertex with the midpoint of the opposite side.

Click and play.

The Orthocenter is the point where the three altitudes of the triangle intersect.

The altitude is a perpendicular line segment from the vertex to the opposite side.

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The circumcenter is equidistant from all three vertices. It lies on the perpendicular bisectors of the three sides. Sometimes it lies outside the triangle. It is also the center of a circle (Circumcircle) that contains the vertices.

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The Incenter is equidistant from all three sides of the triangle. It lies on the angle bisectors. It is also the center of the inscribed circle (Incircle).

Click and play.

Now let’s put all these centers into the same triangle and see how they relate to each other.

Here’s what I found:

• The centers always “line up” in this order: HIGC.
• I and G always remain in the interior of the triangle.
• H and C sometimes lie outside the triangle.
• H is always the closest center to an obtuse angle.
• C is always the farthest center from an obtuse angle.
• In a right triangle, H lies on the vertex and C lies on the midpoint of the opposite side.
• H,G, and C always lie on a straight line.
• The ratio of line segment HG to line segment GC is always 2.
• The ratio of line segment CH to line segment GC is always 3.
• The measure of angle HIG is always greater then 90 degrees.

This one is really great to play with.

We can construct triangles inside triangles. If we connect the midpoints of the sides of a triangle, the result is the medial triangle.

More discoveries:

• The triangles share a centroid.
• The orthocenter of the medial (red) triangle coincides with the circumcenter of the large triangle.
• Each side of the medial triangle is parallel to a side of the larger triangle.

Check it out for yourself.