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Center of Gravity (Posted on 2008-02-06) Difficulty: 3 of 5
Let I, J, K, and L be the incenter and the three excenters of triangle ABC.

What is the center of gravity of these four points?

  Submitted by Bractals    
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Solution: (Hide)
Assume that each of the four points has a mass of one unit.

Procedure:

Replace points I and J with a point EJ having two units of mass and located at the midpoint of line segment IJ. Replace points K and L with a point OJ having two units of mass and located at the midpoint of line segment KL. Replace points EJ and OJ with a point O having four units of mass and located at the midpoint of line segment EJOJ.

If we repeat the above procedure with the letters "JKL" replaced with the letters "KLJ" and then with "LJK", we see that the point O (which is the center of gravity of points I, J, K , and L) is the midpoint of line segments EJOJ, EKOK, and ELOL.

Since I is the orthocenter of triangle JKL, the points EJ, OJ, EK, OK, EL, and OL are six of the nine points which define the nine-point circle for triangle JKL. Therefore, O is the center of the nine-point circle of triangle JKL.

The other three points defining the nine-point circle are the feet of the altitudes of triangle JKL, but those feet are the vertices of triangle ABC. Therefore, the nine-point circle of triangle JKL is the circumcircle of triangle ABC.

Therefore, the center of gravity of points I, J, K, and L is the circumcenter of triangle ABC.

NOTE: Since the four points form an orthocentric system, the above proof is valid nomatter what permutation of the letters I, J, K, and L are applied to the four points.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Puzzle AnswerK Sengupta2022-09-03 12:55:11
SolutionI think I am centered on the solution now...Dej Mar2008-02-08 08:34:51
re: Solution ???Bractals2008-02-07 13:10:29
Some ThoughtsSolution ???Dej Mar2008-02-07 12:10:47
Some Thoughtsre(2): ????Dej Mar2008-02-07 00:32:48
re: ????Charlie2008-02-06 16:42:48
????J2008-02-06 16:20:25
Solutionplaying with Geometer's Sketchpad (spoiler)Charlie2008-02-06 12:45:46
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