## ganeshie8 one year ago 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 :

1. ganeshie8

|dw:1435678656585:dw|

2. ganeshie8

sorry for the bad drawing credit to @dan815

3. dan815

|dw:1435679062232:dw|

4. dan815

are you sure its true, its seems hard to believe

5. ganeshie8

Yes it is true, verified with geogebra

6. dan815

|dw:1435679177952:dw|

7. ganeshie8

\(a\) and \(b\) need to be on the same side of the line

8. ganeshie8

|dw:1435679309628:dw|

9. dan815

oh here is something

10. dan815

we are dealing with 2 sets of parallel slopes here

11. TrojanPoem

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

12. dan815

|dw:1435679496205:dw|

13. TrojanPoem

@ganeshie8 , here is the proof. Get the other part from the last question

14. ganeshie8

that works @TrojanPoem !

15. dan815

trying to not jump into algebra, wanna see if theres a more pure geometry proof

16. ganeshie8

exactly ^ there is a really short proof w/o using trig, and looks you had a good start dan

17. dan815

|dw:1435679888025:dw|

18. ganeshie8
19. TrojanPoem

@ganeshie8 , Like++. Nice animation.

20. dan815

3 sets of similar triangles

21. dan815

|dw:1435680514378:dw|

22. dan815

with sides a b a+b

23. dan815

hey is the motion like a pendulum?

24. dan815

is that a perfect arc

25. TrojanPoem

is think this is a parallelogram with a triangle inscribed

26. TrojanPoem

|dw:1435680716273:dw|

27. dan815
28. TrojanPoem

AC // BD. CD//AB

29. ganeshie8

wow looks like it, nice imagination dan

30. dan815

kind of neat haha, if this relationship defined a pedulum motion in a roundabout way

31. ganeshie8

but it cannot be a circle because the length of that top red segment is changing continuously right

32. dan815

ya thats true, a calculus solution to this might look interesting too, something to look for later

33. ganeshie8

yes @TrojanPoem

34. ganeshie8

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

35. TrojanPoem

You gave up guys ?

36. ganeshie8

nope, we have ur solution so including mine, technically i have 2 solutions... that is far from giving up :D

37. TrojanPoem

Come check the new problem, I am stuck with it.

38. ganeshie8

39. ganeshie8

you're right @dan815 the points do trace circles http://gyazo.com/603ef78cc5a38da14adf431b35db870f

40. TrojanPoem

I did. But i think it's invisible

41. ikram002p

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