Geometrical Situation: Open Discussion.

S-1

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
5 years, 9 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

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 !

- 5 years, 9 months ago

goodnight bro!

- 5 years, 9 months ago

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 .

- 5 years, 9 months ago

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

- 5 years, 9 months ago

I have posted a geometry based solution here.

And thanks for starting this discussion, because I hadn't seen this question. :)

- 5 years, 9 months ago

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 ?

- 5 years, 9 months ago

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,

- 5 years, 9 months ago

But If So then , don't you think angle between Sides should be 120 degree , So that their resultant is minimum ?

- 5 years, 9 months ago

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.

- 5 years, 9 months ago

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

- 5 years, 9 months ago

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?

- 5 years, 9 months ago

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

- 5 years, 9 months ago

Yea, that's fine. Thanks again :)

- 5 years, 9 months ago

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.

- 5 years, 9 months ago

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 ?

- 5 years, 9 months ago

Also, can you state exactly what you want a mathematical example of? And for what stuff? I dint quite get you.

- 5 years, 9 months ago

try this

page 105 and 106

- 5 years, 9 months ago

Thanks for link , I saved that web page and will try it on my study table.

- 5 years, 9 months ago

- 5 years, 9 months ago

Have you tried Paul's Online Notes ?

- 5 years, 9 months ago

Sound's interesting , Could you please tell what was that ? Better if you give link too :)

- 5 years, 9 months ago

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 !!

- 5 years, 9 months ago

This is that problem regarding inequalities right , i too was thinking in the exact manner , how to minimise PA + PB + PC,

- 5 years, 9 months ago

Yes , I still haven't tried it yet but will Lagrange Multipliers do the job ?

- 5 years, 9 months ago

Yes it does.

- 5 years, 9 months ago

Thanks , I'll work on it :)

- 5 years, 9 months ago

I have posted a solution using only the dot product inequality

$(a\theta+b\lambda+c\delta) \le \sqrt {(a^2+b^2+c^2)(\theta^2+\delta^2+\lambda^2)}$

- 5 years, 9 months ago

How do you come up with such new ideas ?

- 5 years, 9 months ago

Its actually a pretty common inequality :)

- 5 years, 9 months ago

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 ?

- 5 years, 9 months ago

thanks for appreciating , but i think its neither just hindsight :)

- 5 years, 9 months ago

:)

- 5 years, 9 months ago

Cauchy Schwartz ?

- 5 years, 9 months ago

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

for any two vectors

$\vec { a } .\vec { b } =\quad \left| \vec { a } \right| \left| \vec { b } \right| cos\theta \quad \le \quad \left| \vec { a } \right| \left| \vec { b } \right| \\ \\ ({ a }_{ 1 }{ b }_{ 1 }+{ a }_{ 2 }{ b }_{ 2 }+{ a }_{ 3 }{ b }_{ 3 })\quad \le \quad \sqrt { { a }_{ 1 }^{ 2 }+{ a }_{ 2 }^{ 2 }+{ a }_{ 3 }^{ 2 } } \sqrt { { b }_{ 1 }^{ 2 }+{ b }_{ 2 }^{ 2 }+{ b }_{ 3 }^{ 2 } }$

- 5 years, 9 months ago

Hmm , I see that you are handy with vectors . So how's your prep for Physics going on ?

- 5 years, 9 months ago

Fine and dandy :)

- 5 years, 9 months ago

Thank you for teaching me how to remember this thing!!!!! I owe you a lot!

- 5 years, 9 months ago

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 !

- 5 years, 9 months ago

I cannot see that! Mentions are invisible to me!

- 5 years, 9 months ago

Check your email id or wherever you registered on Brilliant from . Btw here is the note .

- 5 years, 9 months ago

thanks

- 5 years, 9 months ago

:)

- 5 years, 9 months ago