Where to Learn Barycentric Coordinates

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srisainandan6

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srisainandan6

#1 h Apr 27, 2020, 4:14 PM

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So I was doing this year's AIME #13, and I realized you could solve it with Barycentric Coordinates.

Can anyone post resources/links on where to learn barycentric coordinates? I know mass points but not barycentric coordinates.

Thanks in advance, srisianandan6

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HrishiP

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HrishiP

#2 h Apr 27, 2020, 4:18 PM

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Try EGMO or Evan chen's Barycentric coordinates handout.

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srisainandan6

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srisainandan6

#3 h Apr 27, 2020, 4:29 PM • 2 Y

Y by 554183, Mango247

ok thanks @above

but aren't both of those really advanced. Is there any place easier to learn it

This post has been edited 1 time. Last edited by srisainandan6, Apr 27, 2020, 4:39 PM
Reason: s

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skyscraper

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skyscraper

#4 h Apr 27, 2020, 4:58 PM

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https://www.youtube.com/watch?v=dA7GzG4BIzI

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naman12

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naman12

#5 h Apr 27, 2020, 5:12 PM • 6 Y

Y by skyscraper, TwinPrime, AopsUser101, montana_mathlete, Han1728, math31415926535

I honestly believe that barycentric coordinates is just a matter of memorization. I'll go into a basic intro (that should help you easily understand the rest of EGMO's chapter):

Consider a triangle $\triangle ABC$ (doesn't have to be called $ABC$ ), and we define the homogenized barycentric coordinate as
$P=\left(\dfrac{[PBC]}{[ABC]},\dfrac{[PAC]}{[ABC]},\dfrac{[PAB]}{[ABC]}\right)$ Note that we can define $A=(1,0,0),B=(0,1,0),C=(0,0,1)$ in this system. Furthermore, if a point $P\in BC$ , then we have that $[PBC]=0$ , so thus we easily have that
$P=(0,\text{something},1-\text{something})$ What are the advantages of barycentric coordinates? Here are a few:

We can easily find the area of a triangle $PQR$ as
$[ABC]\begin{vmatrix}p_1&q_1&r_1\\p_2&q_2&r_2\\p_3&q_3&r_3\end{vmatrix}$ where $P=(p_1,p_2,p_3)$ and $Q,R$ are defined analogously.
The formula for many of the standard centers are pretty immediate. For example, consider the centroid. We have that the medians divide a triangle into $6$ triangles of equal areas, so
$G=\left(\dfrac{[GBC]}{[ABC]},\dfrac{[GAC]}{[ABC]},\dfrac{[GAB]}{[ABC]}\right)=\left(\dfrac 13,\dfrac 13,\dfrac 13\right)$ The circumcenter and orthocenter aren't as nice, but still are nice. Furthermore, we have that
$[IBC]=\dfrac 12ar$ and analogous objects, so thus
$I=\left(\dfrac{ar}{2rs},\dfrac{br}{2rs},\dfrac{cr}{2rs}\right)$ where I used $[ABC]=rs$ . However, we can "cancel" the $r$ to get
$I=\left(\dfrac{a}{2s},\dfrac{b}{2s},\dfrac{c}{2s}\right)$
We can actually expand our definition to unhomogenized barycentric coordinates, where we have that
$P=(x:y:z)$ as long as $x,y,z$ are in the same ratio as $[PBC],[PAC],[PAB]$ . That allows us to remove the denominator in, for example, $I$ , so we get
$I=(a:b:c)$ $G=(1:1:1)$
We can actually transform barycentric coordinates into vectors by assuming that
$\vec P=x\vec A+y\vec B+z\vec C\iff P=(x,y,z)$ That helps, for example, with finding the distance between two points, finding perpendicular lines, and also finding circles.

This is a non-exhaustive list. Try to use them for several problems - especially when you think that finding a synthetic solution is going to be pretty hard. I do suggest EGMO for reading however.

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MortemEtInteritum

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MortemEtInteritum

#6 h Apr 27, 2020, 10:28 PM

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srisainandan6 wrote:

ok thanks @above

but aren't both of those really advanced. Is there any place easier to learn it

Yes, but they start from very basics. Barycentric coords is just a rather advanced topic, that doesn't really come up on AIME or AMC.

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CoolCarsOnTheRun

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CoolCarsOnTheRun

#7 h Apr 28, 2020, 12:53 AM • 1 Y

Y by Mango247

A lot of problems can be barybashed, though.

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sub_math

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sub_math

#8 h Apr 28, 2020, 1:20 AM • 1 Y

Y by v4913

wait theoretically if you were to learn bary, could you bary bash all aime geo within a reasonable time limit? is this a good idea?! I hear bary bashing is very powerful but i assume it's mainly for oly >_<

^bc synthetic geo...is not exactly my strong point (aka i have 0 intuition)

This post has been edited 1 time. Last edited by sub_math, Apr 28, 2020, 1:21 AM

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naman12

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naman12

#9 h Apr 28, 2020, 1:25 AM

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sub_math wrote:

wait theoretically if you were to learn bary, could you bary bash all aime geo within a reasonable time limit? is this a good idea?! I hear bary bashing is very powerful but i assume it's mainly for oly >_<

^bc synthetic geo...is not exactly my strong point (aka i have 0 intuition)

No. No. No. Don't try something with like 50 circles. Maybe, but not. Learn all forms of bashing, and come back to learn the ways.

Also, I mainly suggest learning a few important configurations and then bash effortlessly.

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amuthup

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amuthup

#10 h Apr 28, 2020, 1:31 AM • 1 Y

Y by Mango247

When I did the bary chapter in EGMO, I basically skimmed the whole chapter without understanding anything, and then just constantly referred to the areal definition for the problems. Over time, the concepts became much easier to understand, and as @naman12 pointed out, it is just memorization. So don't be scared of Evan Chen's bary handout!

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