Euler Line ProofDate: 11/13/2001 at 23:58:00 From: Natalie Loser Subject: Euler line Prove that if the Euler line of a triangle passes through a vertex, then the triangle is either right or isosceles. Date: 11/14/2001 at 03:05:52 From: Doctor Floor Subject: Re: Euler line Hi, Natalie, Thanks for writing. If (for instance) vertex A lies on the Euler line, then the Euler line of ABC is the line through A and the centroid G = the median through A. So the midpoint of BC is on the line as well. Now if the orthocenter H lies on the median through A, this means that either: - A is not equal to H. Then the median through A is also the altitude from A, and thus ABC is an isosceles triangle with A as top. - A is equal to H. Then A is a right angle, because AB must be the altitude from B to AC, and AC must be the altitude from C to AB. This proves your statement. If you need more help, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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