Incenters, Orthocenters, and the Spieker PointDate: 02/13/2000 at 11:00:16 From: Emily Wilson Subject: Geometry For community service I am working as an assistant teacher to a class of 9th grade geometry honors students in New York. The teacher has asked me to make an extra-credit packet with difficult proofs. I have found some interesting problems, but unfortunately the answers were not provided. My geometry is a little rusty as I took the course two years ago. I would appreciate your help on a problem. Given: M1, M2, M3 are the three midpoints of the sides of triangle ABC, so triangle M1M2M3 is the medial triangle of triangle ABC. Point I is the incenter of triangle ABC and point G is the centroid of triangle ABC. Point S is collinear with I and G as shown in the diagram. 2SG = IG. Prove: S is the incenter of triangle M1M2M3 The book I got this from said a successful completion of the proof would give you that the points I, G and S are collinear in any triangle and that IG and GS are in a ratio of 2:1. It also said that point S is sometimes called the Spieker point. Thanks, Emily Date: 02/13/2000 at 13:55:37 From: Doctor Floor Subject: Re: Geometry Hi Emily, Thanks for your question. If you are to present the problem, you should be very clear that S is on the other side of G from I. If that is clear, there is not much to prove really. The result follows from the observation that G is the center of similitude of ABC and its medial triangle. Thus we conclude from AG:GM1 = 2:1 that if we take a point X in triangle ABC and the corresponding point Y in triangle M1M2M3, then G is between X and Y and XG:GY = 2:1. So when we take X is the incenter, then Y is the incenter of M1M2M3. This point S - the Spieker point - is a remarkable point. It is the center of mass of the perimeter of triangle ABC. This can be seen in the following way: Let a = |BC|, b = |AC| and c = |AB|. We can represent the masses of the three sides of ABC by placing weights of magnitudes a on M1, b on M2 and c on M3. Let Z be the point that divides segment M2M3 in ratio M2Z:ZM3 = c:b (so Z is the center of mass of the two weights on M2 and M3), then we can replace the weights of M2 and M3 by a weight of magnitude b+c on Z. Now the center of mass of the weights on M1 and Z, and thus the center of mass of the perimeter, is the point T dividing segment M1Z in ratio M1T:TZ = a:b+c. From this we can see that area of triangle M2M3T is a/(a+b+c) times the area of triangle M1M2M3. In the same way we can see that the area of triangles M1M3T and M1M2T are b/(a+b+c) and c/(a+b+c) times the area of triangle M1M2M3 respectively. Using that the sides of M1M2M3 are of half the lengths of the sides of ABC and that area(M2M3T):area(M1M3T):area(M1M2T) = a:b:c, we can conclude that the altitudes from T to the sides of triangle M1M2M3 are all three of equal length. And thus T is the incenter of M1M2M3, the Spieker point. If you have more questions, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ Date: 02/13/2000 at 14:55:07 From: Emily Wilson Subject: Re: Geometry I don't really remember what the center of similitude is and I'm pretty sure my students haven't gotten to that yet. What is it and is there another way to prove that S is the incenter of triangle M1M2M3? Date: 02/15/2000 at 02:26:56 From: Doctor Floor Subject: Re: Geometry Hi, Emiliy, Thanks for responding. Take a plane geometric figure and a fixed point P. When, from this point P, you multiply the distances to the points of the figure with a fixed number (negative -> take the opposite side of P) you get a figure similar to the original figure. When of two similar figures such a point exists, which is the case for a triangle and its medial triangle, then the point is called sometimes "Center of Similitude" or "Center of Multiplication." I hope that clears things up. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ Date: 02/13/2000 at 16:20:17 From: Emily Wilson Subject: Re: Geometry Hi, this is Emily again. When you bisect the exterior angles of a triangle, they intersect at three points, called the excenters. The incenter of the triangle is the orthocenter of the triangle formed by the 3 excenters. How do I prove this? (What tells me that the angle bisectors of the smaller triangle are also the altitudes of the larger one?) Does it have something to do with drawing the circumcircle around the smaller triangle? Date: 02/15/2000 at 03:28:00 From: Doctor Floor Subject: Re: Geometry Hi, Emily, Thanks for your new question. There are two observations you should make to prove this: 1. The internal and external bisectors from one vertex are perpendicular to each other. This can be seen in the following way: When the internal bisector bisects an angle of magnitude A degrees, then the external bisector bisects an angle of magnitude 180 - A degrees. So the bisected angles are A/2 and 90-A/2 degrees, which makes 90 degrees together. 2. The point of intersection of two external bisectors is on the internal bisector of the third angle (this can be shown by reasoning that the angle bisectors are lines of points equidistant from the sidelines of the triangle, and seeing in what regions the internal and external bisectors are). When you draw this, you see that this gives that the incenter is the orthocenter of the excenters If you need more help, just write back. Kind regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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