Euler's Nine-point CircleDate: 02/21/99 at 23:59:43 From: asit bhattacharyya Subject: Nine-point Circle Could you give me some information on "nine-point circle"? Thanks. Date: 02/28/99 at 06:18:05 From: Doctor Floor Subject: Re: Nine-point Circle This is a very famous triangle problem that was presented by the Swiss mathematician Euler in 1765. Let ABC be a triangle. Let A~ be the midpoint of BC, B~ of AC, and C~ of AB. Also let A' be the foot of the altitude from A on BC, and let B' and C' be the feet of the other two altitudes. The three altitudes intersect at a common point, the orthocenter H. Finally, let A' be the midpoint of AH, B" of BH and C" of CH. The nine-point circle is the circle passing through the nine points A~, B~, C~, A', B', C', A", B" and C": Two arguments are needed to prove that these nine points are indeed concyclic: 1. First consider quadrilateral A'A~B~C~. This is a trapezoid, because A'A~ is parallel to B~C~. Note that A'C~ = AC~ = AB/2 and that A~B~ = AB/2 too, so A'A~B~C~ is an isosceles trapezoid. An isosceles trapezoid can be circumscribed by a circle, so the circle circumscribing A~B~C~ passes through A'. In the same way, it passes through B' and C'. 2. Observe that C~A" is parallel to BH and thus parallel to BB' too. Also C~A~ is parallel to AC. But this gives us that angle A~C~A" is right. Since angle A"A'A~ is right too, we know that the circle with diameter A"A~ passes through both C~ and A'. This must be the same circle as in our first argument, because this circle passes through A~, C~, and A'. So our circle from argument 1. passes through A", and in the same way through B" and C". We can see that triangle A"B"C" is the product figure of ABC with factor 0.5 over H. This means that the center of the nine point circle N is the midpoint of the circumcenter and the orthocenter. This also means that N must be on the Euler line. You can read about the Euler line in our archives: Euler Line http://mathforum.org/dr.math/problems/christen6.8.98.html A very nice result on the nine point circle is proven by Feuerbach in 1822: it is tangent to the incircle and the excircles of a triangle. The center of the nine point circle, also called the nine point center, is a very well-known triangle center. A lot of triangle centers are presented in a site by Professor Clark Kimberling of the University of Evansville: Triangle Centers http://cedar.evansville.edu/~ck6/tcenters/ I hope this helps! Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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