Trouble with Maths – Study Abroad

oLahav said – Thu, 05 Mar 2009 16:01:30 -0000 ( Flag Edit Mark / Unmark Best Link )

I’ll start with the definitions:

The centroid of the triangle is the point of intersection between the 3 lines from each vertex to the center of the opposite side (cutting the side in 1/2). The orthocenter is the point of intersection between the 3 altitudes of the triangle (lines from a vertex that’s perpendicular to the opposite side). The incenter is the point of intersection between the 3 angle bisectors.

It’s pretty easy to see that the centroid and the incenter are generally closer together than the centroid and the orthocenter, simply because both the centroid and the incenter my always fall within the triangle, but the orthocenter doesn’t, simple by construction of the triangle.

For a fuller explanation… it’s too complicated to just post in a discussion, but I’ll try to cover the basics. The centroid and the orthocenter both lie on the same line as the center of the nine-point circle, which is a circle that passes through all the midpoints of each side, all of bases of the altitudes, and all the midpoints of the portion of each altitude that starts at the vertex and ends at the orthocenter (this circle can be constructed in every triangle. Picture here). The interesting property of this circle is that the center lies at the midpoint between the centroid and the circumcenter (the center of a circle inscribing the triangle). The radius of the nine-point circle is half that of the circumcircle.

The incenter can also be thought of as the center of the largest circle that can be inscribed within a triangle. It’s tangent to the 3 excircles (with the point of contact lying on the triangle’s respective side) and is also tangent at one point, called the Feuerbach Point, to the 9-point circle. The incenter forms an orthocentric system with the centres of the 3 excircles, which is a 4-point system in which all 4 possible triangles share the same 9-point circle. It follows that the centroid will lie at the centre of this 9-point circle, which from above will be at twice the distance from the original orthocenter.

I know it’s all terribly unclear, maybe drawing it out will help. The bottom line is, the centroid is twice as far from the orthocenter as it is from the incenter because math says so.

P.S. is it just me, or are there way too many circles involved in an explanation about triangles?

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