Where is Fletcher?
Triangle geometry has identified over 50,000 triangle centers. There is even an enclopedia of them. Some are well known like centroid, incircle, and circumcenter. But most are obscure. Of the obscure, my personal favorite is the Fletcher point. It lies at the intersection of the Gergonne line and the Soddy line. As an added attraction, these two lines (see link) happen to be perpendicular!
Given a 3-4-5 right triangle located on the cartesian plane at ( 0 , 0 ) , ( 3 , 0 ) , ( 0 , 4 ) , in which quadrant does its Fletcher point lie?
Bonus: what is your favorite point?
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2 solutions
Doing the calculations, Fletcher point of Fletcher's triangle is ( − 1 3 9 , − 1 3 2 0 ) .
Oops, I meant to write this as a comment at Fletcher's solution. Now, I guess I have to write a complete solution :)
A complete solution? Oh no, I wouldn't wish that on anyone. That's why I just drew a picture!
I agree with these coordinates for Fletcher's points. Since the incentre is ( 1 , 1 ) , and the Gergonne point is easy to find (as the intersection of the lines x + 4 y = 1 and 3 x + y = 1 ), the equation for the Soddy line (which passes through the incentre and the Gergonne point) is pretty easy. Since the Gergonne line is perpendicular to the Soddy line, and passes through ( 9 , − 8 ) , the intersection of 3 x + 4 y = 1 and x + y = 1 , it is not that hard to find the Fletcher point.
That's true enough. However, a "complete solution" could include the definition and construction of the other points in the figure which define the Gergonne and Soddy lines. At least that's what I had in mind when I made the (somewhat whimsical) comment to Thanos. In retrospect, I could have been more clear about this.
@Mark Hennings , @Fletcher Mattox - Ok, ok! Anyway, now that Mark outlined the process, I am no longer "obliged" to write a "complete solution". Fletcher's links are more than enough. By the way, why isn't there any Thanos , or Mark point in the list? this is not fair...
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Harder to find Mark or Thanos as surnames (except in the Marvel universe, perhaps)?
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I think Thanos is a popular Greek name. Could be shortened from Thanatos , the greek god of death.
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@A Former Brilliant Member – It is true that "thanatos" means death, but my name is the exact opposite. "Thanos" is a diminutive of "Athanassios". The noun "athanassia" means immortality. I'm not a supervillain either :)
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@Thanos Petropoulos – Cool. I'm Greek too, my name's Perseus(yes, Perseus Jackson, Son of Poseidon).
For this triangle, the Fletcher point is in quadrant III.