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 Zelaron Gaming Forum Determining if a complex number w = f(z1, z2, z3) is a known triangle center

 Determining if a complex number w = f(z1, z2, z3) is a known triangle center
 Posted 2018-11-23, 08:53 AM I've found a number of closely related functions, each of which takes three complex numbers $\reverse z_1,\,z_2$, and $\reverse z_3$ (which we can consider as the vertices of a triangle) as its arguments, and outputs a complex number $\reverse w$. A few of these functions represent well-known triangle centers $\reverse w$, such as the centroid (that is, $\reverse w=(z_1+z_2+z_3)/3)$), or the zeros and critical points of the polynomial $\reverse R(z) = (z-z_1)(z-z_2)(z-z_3)$. Other functions among them are less obvious, such as the following point $\reverse w=p$: $\reverse p = \frac{(z_2+z_3) z_1^2+\left(z_2^2-6 z_3 z_2+z_3^2\right) z_1+z_2 z_3 (z_2+z_3) \sqrt{-3(z_1-z_2)^2 (z_1-z_3)^2 (z_2-z_3)^2}}{2 \left(z_1^2+z_2^2+z_3^2-z_1z_2 - z_1z_3-z_2z_3\right)}$ As far as I can tell, by comparing the point $\reverse p$ to different triangle centers in GeoGebra Classic 6, it seems to be the first isodynamic point to a high degree of precision for all triangles that I've thrown at it so far. My problem is twofold: $\reverse (1)$ How do I prove that $\reverse p$ is (or isn't) the first isodynamic point of the triangle $\reverse (z_1,\,z_2,\,z_3)$? $\reverse (2)$ How do I (numerically) match $\reverse w$ with likely triangle centers based on the Encyclopedia of Triangle Centers? GeoGebra uses a small portion of the material in the ETC, but unfortunately, it fails to implement some important triangle centers with low indices. Suggestions for other, similar resources are welcome, too. As for $\reverse (1)$, I think a viable starting point might be to convert $\reverse p$ into barycentric or normalized trilinear coordinates and directly compare the result to the coordinates given in the ETC, but I'm not sure how to do this. It seems to be slightly easier to use Cartesian coordinates as a starting point, but Mathematica (for example) generally doesn't seem to be very inclined to return the real and imaginary parts of $\reverse p$ (for general $\reverse z_k$, e.g. when you let $\reverse z_k = a_k + i b_k$ for real numbers $\reverse a_k,\,b_k,\,k=1,2,3,$ and simplify the expressions). Regarding $\reverse (2)$, the following website can supposedly be used to compare points to almost all entries in the ETC: https://faculty.evansville.edu/ck6/e...ch_6_9_13.html Unfortunately, I seem to get inconsistent results from my attempts to implement the algorithm on the website above in Mathematica. More specifically, certain triangle centers that I've tested (such as the centroid) yield the correct coordinates in the list, while other obvious ones that I've tested are either not listed, or have the wrong index. I suspect that I've misunderstood the information on the website in some elementary way, so feel free to correct my algorithm (for $\reverse p$ above, in the example that follows): $\reverse (i)$ Choose $\reverse z_1,\,z_2,\,z_3$ as the vertices of a triangle with side lengths $\reverse 6,\,9,$ and $\reverse 13$, e.g. $\reverse z_1 = 0,\,z_2 = 9,$ and $\reverse z_3 = (-26/9) + (8/9)\sqrt{35}i$. $\reverse (ii)$ Solve the following linear system of equations for $\reverse u,v,w$: $\reverse u\,\text{Re}(z_1) + v\,\text{Re}(z_2) + w\,\text{Re}(z_3) = \text{Re}(p)$ $\reverse u\,\text{Im}(z_1) + v\,\text{Im}(z_2) + w\,\text{Im}(z_3) = \text{Im}(p)$ $\reverse u + v + w = 1,$ where $\reverse u:v:w$ are the barycentric coordinates for $\reverse p$. (Here, $\reverse p$ can be approximated to a high degree of precision, e.g. to avoid problems with Mathematica...) $\reverse (iii)$ Let $\reverse a=6,\,b=9$, and $\reverse c=13$ (or, more ideally, just use the Pythagorean theorem...), and define $\reverse x = u/a,\,y = v/b,\,z = w/c$. Furthermore, calculate the area of the triangle as $\reverse A = 4\sqrt{35}$. $\reverse (iv)$ Calculate $\reverse kx = 2Ax/(ax+by+cz) = 2Ax/(u+v+w)$, which should be the sought "coordinate" in the table. Carrying out the algorithm above yields $\reverse kx \approx 0.14368543660$, whereas the coordinate for the first isodynamic point (with index $\reverse 15$ in the table) is $\reverse \sim 3.10244402065$. Perhaps $\reverse p$ isn't actually the first isodynamic point, but I like to think that I've done something wrong... "Stephen Wolfram is the creator of Mathematica and is widely regarded as the most important innovator in scientific and technical computing today." - Stephen Wolfram Last edited by Chruser; 2018-11-23 at 11:11 AM.
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Chruser

 Posted 2018-11-23, 02:53 PM in reply to Chruser's post "Determining if a complex number w =..." Why do you do this
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-Spector-

Posted 2018-11-23, 10:11 PM in reply to -Spector-'s post starting "Why do you do this"
-Spector- said: [Goto]
 Why do you do this

Some people just don't understand the dangers of indiscriminate surveillance.
"Stephen Wolfram is the creator of Mathematica and is widely regarded as the most important innovator in scientific and technical computing today." - Stephen Wolfram
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Chruser

 Posted 2018-12-02, 09:15 AM in reply to Chruser's post starting "Some people just don't understand the..." I discovered this the other day: 9 squared = 81, 9% of 9 is .81 8 squared = 64, 8% of 8 i .64 7 squared = 49, 7% of 7 is .49 6 squared = 36, 6% of 6 is .36 .... So there's that [Fixed my fuck up - thanks for outlining my flaws, K_A (dick)] Last edited by -Spector-; 2018-12-03 at 03:50 AM.
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-Spector-

Posted 2018-12-02, 05:26 PM in reply to -Spector-'s post starting "I discovered this the other day: 9..."
-Spector- said:
 6 squared = 36, 6% of 36 is .36

Anyway, the pattern you are displaying is a fairly obvious one. 9^2=9*9=81 whereas 9%*9=0.09*9=0.81. 0.09 is two orders of magnitude smaller than 9, which is why the answer also ends up being two orders of magnitude smaller (81 vs 0.81).
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 !King_Amazon!

 Posted 2018-12-02, 06:05 PM in reply to !King_Amazon!'s post starting "Check your math, buddy. Anyway, the..." Oh I realize how basic it is, its just fun to find patterns. Can you point out which part of my math I need to check? When I do 6^2 I get 36. When I do 6 * .06 I get .36
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-Spector-

Posted 2018-12-02, 08:35 PM in reply to -Spector-'s post starting "Oh I realize how basic it is, its just..."
-Spector- said: [Goto]
 Oh I realize how basic it is, its just fun to find patterns. Can you point out which part of my math I need to check? When I do 6^2 I get 36. When I do 6 * .06 I get .36
That is correct, but not what you wrote above.

Quote:
 6 squared = 36, 6% of 36 is .36
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 !King_Amazon!

 Posted 2018-12-03, 03:47 AM in reply to !King_Amazon!'s post starting "That is correct, but not what you wrote..." Fuck .
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-Spector-

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