show that \(a+b=c\) where \(a,b,c\) are the perpendicular distances from vertices to a line passing through the centroid of an equilateral triangle as shown :

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show that \(a+b=c\) where \(a,b,c\) are the perpendicular distances from vertices to a line passing through the centroid of an equilateral triangle as shown :

Mathematics
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sorry for the bad drawing credit to @dan815
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are you sure its true, its seems hard to believe
Yes it is true, verified with geogebra
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\(a\) and \(b\) need to be on the same side of the line
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oh here is something
we are dealing with 2 sets of parallel slopes here
Well, c = y sinw , a = y sin ( w - 60) , b = y sin(w +60 ) we have to prove that a + b = c left: a + b = ysin(w-60) + ysin(w+60) = y( sin(w-60)+ sin(w+60)) = y[ sinwcos60 - coswsin60 + sinwcos60 + coswsin60] = y* 2sinwcos60 = ysinw = c
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@ganeshie8 , here is the proof. Get the other part from the last question
that works @TrojanPoem !
trying to not jump into algebra, wanna see if theres a more pure geometry proof
exactly ^ there is a really short proof w/o using trig, and looks you had a good start dan
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http://assets.openstudy.com/updates/attachments/558b8754e4b075714118a70e-ganeshie8-1435679847507-zzz.gif
@ganeshie8 , Like++. Nice animation.
3 sets of similar triangles
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with sides a b a+b
hey is the motion like a pendulum?
is that a perfect arc
is think this is a parallelogram with a triangle inscribed
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http://prntscr.com/7n5vrt
AC // BD. CD//AB
wow looks like it, nice imagination dan
kind of neat haha, if this relationship defined a pedulum motion in a roundabout way
but it cannot be a circle because the length of that top red segment is changing continuously right
ya thats true, a calculus solution to this might look interesting too, something to look for later
let me know if you want me to share my solution because i see now it wont spoil anything as there can be multiple solutions
You gave up guys ?
nope, we have ur solution so including mine, technically i have 2 solutions... that is far from giving up :D
Come check the new problem, I am stuck with it.
have you posted it already
you're right @dan815 the points do trace circles http://gyazo.com/603ef78cc5a38da14adf431b35db870f
I did. But i think it's invisible
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