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Geometrical Situation: Open Discussion.

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Problem: Find Minimum value of PA+PB+PC , which have constrained that a+b+c is constant integer (for Simplicity let it be ' 5 ' )

I want only geometric approach . This is open discussion , if you are able to get some steps then share it with us , we all brilliantian's together try to complete that.

Note by Deepanshu Gupta
2 years, 2 months ago

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Guys, I think there is a very simple way to solve this problem.... I have that feeling, check out my solution to see if it is right here

@Deepanshu Gupta @Mvs Saketh Raghav Vaidyanathan · 2 years, 2 months ago

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I have posted a geometry based solution here.

And thanks for starting this discussion, because I hadn't seen this question. :) Shashwat Shukla · 2 years, 2 months ago

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@Shashwat Shukla Hey I have one doubt , If I treat this side as Force Vector's and Try to minimize the restultant force then what will we get ? What Do you Think about it ? May be this not fit in this situation , do we make a suitable situation for using this concept ? I'am really Interested in making an situation in which we can analyse Such Minimum value of expresion using Force concept. I'am not able to come up with particular conclusion about such situation individually . Could You Please Help me out . This may sound silly Bit , but it will be great if we do some discusion on it @Shashwat Shukla and all other guy's what do you think ? @Mvs Saketh what do you think ? Deepanshu Gupta · 2 years, 2 months ago

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@Deepanshu Gupta The truth is @Deepanshu Gupta , that is almost exactly the solution, that is the solution Raghav posted, however it need not be force, it can be any vectors, infact the same inequatlity u used to solve the "is it too complex" is more or less applied here, Mvs Saketh · 2 years, 2 months ago

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@Mvs Saketh But If So then , don't you think angle between Sides should be 120 degree , So that their resultant is minimum ? Deepanshu Gupta · 2 years, 2 months ago

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@Deepanshu Gupta That isn't always necessary: If we have vectors of magnitude 1,2,3 then the minimum magnitude of force is zero when the angle is 180. Shashwat Shukla · 2 years, 2 months ago

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@Shashwat Shukla ohh Yes , I'am getting Confused with Fermat Point theoram , which i had seen recently on net , i understood very little about it , due to my weak litrature skills , But i was thinking that in fermat's point angle was 120degree to get minimum sum , then what should be reason behind that ? actually I solved 2-3 questions on brilliant earlier using my own doubtfull concept that Sum of three sides is minimum when angle b/w them is 120 degree. But i never come up with proof of such . But Yes you are right that 1,2,3 makes min. at 180 degree. Also Can You Think of such sitution where this can fitt ? I have studied in a solution of brilliant question , that People are using Surface Tension concept in Solving Minimisation problems of sum of Sides etc. Do you Know about it ? I seriously wanted to learn this technique Deepanshu Gupta · 2 years, 2 months ago

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@Deepanshu Gupta I've actually done the surface tension experiment in real life! :D...Take a wooden board and fix some 5 or 6 pins on it and dip it in soap solution...The soap will connect the dots such that the path has the least length. It actually works!! As you know, this is due to minimum energy of this configuration...

But I haven't seen such a problem on brilliant... Which one are you talking about? Shashwat Shukla · 2 years, 2 months ago

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@Shashwat Shukla yes , that's really Interesting phenomena , I loved to study about it . But Actually It's a long time ago I see that question , so I don't remember that question yet , But I'am saerching it if I got , then I will give you link for that Deepanshu Gupta · 2 years, 2 months ago

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@Deepanshu Gupta Yea, that's fine. Thanks again :) Shashwat Shukla · 2 years, 2 months ago

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@Deepanshu Gupta Ah, I see you know this... What actually happens there is that the liquid tries to minimize it's surface area. Since for constant width, the surface area is directly proportional to the length, the liquid takes the shape of the fermat point connected to the vertices of the triangle. Raghav Vaidyanathan · 2 years, 2 months ago

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@Raghav Vaidyanathan Yes , Could You state an mathematical example for it ? Also do you know any good Link(I mean which is , in Easy language) For such stuffs ? Deepanshu Gupta · 2 years, 2 months ago

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@Deepanshu Gupta Also, can you state exactly what you want a mathematical example of? And for what stuff? I dint quite get you. Raghav Vaidyanathan · 2 years, 2 months ago

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@Deepanshu Gupta try this

page 105 and 106 Raghav Vaidyanathan · 2 years, 2 months ago

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@Raghav Vaidyanathan Thanks for link , I saved that web page and will try it on my study table. Deepanshu Gupta · 2 years, 2 months ago

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@Deepanshu Gupta Could you please give the link about the problem that you were talking about? Shashwat Shukla · 2 years, 2 months ago

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@Deepanshu Gupta Have you tried Paul's Online Notes ? Azhaghu Roopesh M · 2 years, 2 months ago

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@Azhaghu Roopesh M Sound's interesting , Could you please tell what was that ? Better if you give link too :) Deepanshu Gupta · 2 years, 2 months ago

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@Deepanshu Gupta Here's the link . It contains info on roughly all the topics in Maths . But you'll have to search for it !! Check it out !! Azhaghu Roopesh M · 2 years, 2 months ago

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Hey thanks guy's and sorry , I got too many interesting stuff's so I'am going on my study table since I'am too excited and wanted to analyse them. I will report you back after doing some work on them , till than thanks alot for help. Good Night Brothers ! Deepanshu Gupta · 2 years, 2 months ago

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@Deepanshu Gupta goodnight bro! Raghav Vaidyanathan · 2 years, 2 months ago

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Thanks all Guy's , Sorry for late reply , actually My BSNL Broadband Internet become dad Last Night , I really get offended from this broadband connection. But Any way Thanks For this discussion and sharing Your Ideas related to that question . Deepanshu Gupta · 2 years, 2 months ago

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This is that problem regarding inequalities right , i too was thinking in the exact manner , how to minimise PA + PB + PC, Mvs Saketh · 2 years, 2 months ago

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@Mvs Saketh Yes , I still haven't tried it yet but will Lagrange Multipliers do the job ? Azhaghu Roopesh M · 2 years, 2 months ago

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@Azhaghu Roopesh M Yes it does. Shashwat Shukla · 2 years, 2 months ago

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@Shashwat Shukla Thanks , I'll work on it :) Azhaghu Roopesh M · 2 years, 2 months ago

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@Azhaghu Roopesh M I have posted a solution using only the dot product inequality

\( (a\theta+b\lambda+c\delta) \le \sqrt {(a^2+b^2+c^2)(\theta^2+\delta^2+\lambda^2)}\) Mvs Saketh · 2 years, 2 months ago

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@Mvs Saketh How do you come up with such new ideas ? Azhaghu Roopesh M · 2 years, 2 months ago

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@Azhaghu Roopesh M Its actually a pretty common inequality :) Mvs Saketh · 2 years, 2 months ago

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@Mvs Saketh I wasn't referring to the inequality , I was asking how do you connect some totally diff concepts and use them to solve questions . Experience or practice ? Azhaghu Roopesh M · 2 years, 2 months ago

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@Azhaghu Roopesh M thanks for appreciating , but i think its neither just hindsight :) Mvs Saketh · 2 years, 2 months ago

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@Mvs Saketh :) Azhaghu Roopesh M · 2 years, 2 months ago

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@Mvs Saketh Cauchy Schwartz ? Azhaghu Roopesh M · 2 years, 2 months ago

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@Azhaghu Roopesh M Yes, i call it dot product inequality, because its hard to remember cauchy schwartz and also because

for any two vectors

\(\vec { a } .\vec { b } =\quad \left| \vec { a } \right| \left| \vec { b } \right| cos\theta \quad \le \quad \left| \vec { a } \right| \left| \vec { b } \right| \\ \\ ({ a }_{ 1 }{ b }_{ 1 }+{ a }_{ 2 }{ b }_{ 2 }+{ a }_{ 3 }{ b }_{ 3 })\quad \le \quad \sqrt { { a }_{ 1 }^{ 2 }+{ a }_{ 2 }^{ 2 }+{ a }_{ 3 }^{ 2 } } \sqrt { { b }_{ 1 }^{ 2 }+{ b }_{ 2 }^{ 2 }+{ b }_{ 3 }^{ 2 } } \) Mvs Saketh · 2 years, 2 months ago

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@Mvs Saketh Thank you for teaching me how to remember this thing!!!!! I owe you a lot! Raghav Vaidyanathan · 2 years, 2 months ago

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@Raghav Vaidyanathan Raghav , Vraj just mentioned you in a note of his , did you notice ?He's been trying to get you there for a while now ! Azhaghu Roopesh M · 2 years, 2 months ago

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@Azhaghu Roopesh M I cannot see that! Mentions are invisible to me!

I dont get notifications! Raghav Vaidyanathan · 2 years, 2 months ago

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@Raghav Vaidyanathan Check your email id or wherever you registered on Brilliant from . Btw here is the note . Azhaghu Roopesh M · 2 years, 2 months ago

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@Azhaghu Roopesh M thanks Raghav Vaidyanathan · 2 years, 2 months ago

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@Raghav Vaidyanathan :) Azhaghu Roopesh M · 2 years, 2 months ago

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@Mvs Saketh Hmm , I see that you are handy with vectors . So how's your prep for Physics going on ? Azhaghu Roopesh M · 2 years, 2 months ago

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@Azhaghu Roopesh M Fine and dandy :) Mvs Saketh · 2 years, 2 months ago

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