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Friend of mine saw very interesting triangle centers which are closely located for any triangle. Those are Hofstadter point D (X360: https://en.wikipedia.org/wiki/Hofstadter_points) and the Outer Soddy center E (X176: http://mathworld.wolfram.com/OuterSoddyCenter.html).

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The question I have a bit out of scope may be: But can this cause any fundamental physical law or some formula in discrete mathematics? In other words I would like to ask about the applications of this.

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And even more if we consider the line connecting Hofstadter 0 and 1 points (X359 and X360) the X176 is located amazingly close to this line. The maximum distance I was able to get is less than 0.05 for triangle with area grater than 15000.

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  • $\begingroup$ Please just pay an attention that X360 is a transcendental point while X176 is not. $\endgroup$ – Gevorg Hmayakyan Dec 1 '17 at 5:50
  • $\begingroup$ As the Encyclopedia of Triangle centers lists 15423 different centers, it is not a surprise if two of them happen to lie one near to the other. $\endgroup$ – Aretino Dec 1 '17 at 11:48
  • $\begingroup$ I agree. But those two are extremely close. And those are interesting because of one is transcendental and another is not. For example for the triangle with area 205 the maximum distance I was able to find is 0.47. $\endgroup$ – Gevorg Hmayakyan Dec 1 '17 at 19:07
  • $\begingroup$ The Geogebra has a special function trianglecenter(A,B,C,<point number>). $\endgroup$ – Gevorg Hmayakyan Dec 2 '17 at 10:12

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