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Pythagoras Point :: Challenge

Consider an Triangle ABC and "P" Point inside this Triangle such that : \(PA=3\) and \(PB=4\) and \(PC=5\) . Then Find Maximum Possible Value of : \(\displaystyle{{ ({ AB }^{ 2 }+{ BC }^{ 2 }+{ CA }^{ 2 }) }_{ max }}\)


I got This Interesting situation from my friend ,But I couldn't able to solve it . Can You ? Take it as Challenge

Thanks!

Note by Karan Shekhawat
2 years, 2 months ago

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Okay , I done it in This way : Let Consider our Triangle frame in Argand Plane By assuming \(P(0)\quad ,\quad A({ Z }_{ 1 })\quad ,\quad B({ Z }_{ 2 })\quad \& \quad C({ Z }_{ 3 })\) So our Task is to find \({ E }_{ max }\)

Note : Here I will repeatedly Use : The Standard Properties : \(\bullet { \left| { Z } \right| }^{ 2 }={ (Z) }(\overset { \_ }{ { Z } } )\\ \bullet Z+\overset { \_ }{ { Z } } =2Re(Z)\)

\(\displaystyle{E={ \left| { Z }_{ 1 }-{ Z }_{ 2 } \right| }^{ 2 }+{ \left| { Z }_{ 2 }-{ Z }_{ 3 } \right| }^{ 2 }+{ \left| { Z }_{ 3 }-{ Z }_{ 1 } \right| }^{ 2 }\\ E=({ Z }_{ 1 }-{ Z }_{ 2 })(\overset { \_ }{ { Z }_{ 1 } } -\overset { \_ }{ { Z }_{ 2 } } )+({ Z }_{ 2 }-{ Z }_{ 3 })(\overset { \_ }{ { Z }_{ 2 } } -\overset { \_ }{ { Z }_{ 3 } } )+({ Z }_{ 3 }-{ Z }_{ 1 })(\overset { \_ }{ { Z }_{ 3 } } -\overset { \_ }{ { Z }_{ 1 } } )\\ E=2({ \left| { Z }_{ 1 } \right| }^{ 2 }+{ \left| { Z }_{ 2 } \right| }^{ 2 }+{ \left| { Z }_{ 3 } \right| }^{ 2 })-2Re({ Z }_{ 1 }\overset { \_ }{ { Z }_{ 2 } } +{ Z }_{ 2 }\overset { \_ }{ { Z }_{ 3 } } +{ Z }_{ 3 }\overset { \_ }{ { Z }_{ 1 } } )\quad \quad .\quad .\quad .\quad (1)}\)

Now Consider \(\displaystyle{{ \left| { Z }_{ 1 }{ +Z }_{ 2 }+{ Z }_{ 3 } \right| }^{ 2 }\quad \ge 0\\ ({ Z }_{ 1 }{ +Z }_{ 2 }+{ Z }_{ 3 })(\overset { \_ }{ { Z }_{ 1 } } +\overset { \_ }{ { Z }_{ 2 } } +\overset { \_ }{ { Z }_{ 3 } } )\ge 0\\ { \left| { Z }_{ 1 } \right| }^{ 2 }+{ \left| { Z }_{ 2 } \right| }^{ 2 }+{ \left| { Z }_{ 3 } \right| }^{ 2 }+2Re({ Z }_{ 1 }\overset { \_ }{ { Z }_{ 2 } } +{ Z }_{ 2 }\overset { \_ }{ { Z }_{ 3 } } +{ Z }_{ 3 }\overset { \_ }{ { Z }_{ 1 } } )\ge 0\\ { (2Re({ Z }_{ 1 }\overset { \_ }{ { Z }_{ 2 } } +{ Z }_{ 2 }\overset { \_ }{ { Z }_{ 3 } } +{ Z }_{ 3 }\overset { \_ }{ { Z }_{ 1 } } )) }_{ min }=-({ \left| { Z }_{ 1 } \right| }^{ 2 }+{ \left| { Z }_{ 2 } \right| }^{ 2 }+{ \left| { Z }_{ 3 } \right| }^{ 2 })\quad .\quad .\quad .\quad (2)\\ \\ { E }_{ max }=3({ \left| { Z }_{ 1 } \right| }^{ 2 }+{ \left| { Z }_{ 2 } \right| }^{ 2 }+{ \left| { Z }_{ 3 } \right| }^{ 2 })\\ { E }_{ max }=3({ 3 }^{ 2 }+{ 4 }^{ 2 }+{ 5 }^{ 2 })\\ \boxed { { E }_{ max }=150 } }\)

But I Didn't able to find AB , BC , AC as individual at this condition ! Deepanshu Gupta · 2 years, 2 months ago

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@Deepanshu Gupta nice Omar El Mokhtar · 2 years ago

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@Deepanshu Gupta This is awesome ! Karan Shekhawat · 2 years, 2 months ago

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@Karan Shekhawat How did you wrote in a white box? Megh Choksi · 2 years, 2 months ago

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@Megh Choksi I Don't Know , Beacuse I didn't Do that Karan Shekhawat · 2 years, 2 months ago

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Interesting situation ! I'am getting 150 is it correct ? I used Complex numbers @KARAN SHEKHAWAT Deepanshu Gupta · 2 years, 2 months ago

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@Deepanshu Gupta I get the same result, with \(AB = 5, BC = \sqrt{73}\) and \(AC = \sqrt{52}\). Brian Charlesworth · 2 years, 2 months ago

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@Brian Charlesworth How did You get Individual Values of all sides ? Can You Please show this ? Because I'am able To find Maximum Value , But I'am unable To find What are Individual Values of Sides . Thanks! Deepanshu Gupta · 2 years, 2 months ago

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@Deepanshu Gupta Your solution is far more elegant than mine. My approach was to first let the "central" angles at \(P\) opposite sides \(AB, BC,CA\) be \(\alpha, \beta, \gamma\), respectively. I then used the Cosine Law to establish the equation

\(AB^{2} + BC^{2} + CA^{2} = 100 - 24\cos(\alpha) - 40\cos(\beta) - 30\cos(\gamma) = f(\alpha, \beta, \gamma)\).

I then applied Lagrange multipliers under the condition \(g(\alpha, \beta, \gamma) = \alpha + \beta + \gamma = 2\pi\). Noting the negative signs in the expression for \(f(\alpha, \beta, \gamma)\) I further applied the condition that each of the three angles were \(\ge \frac{\pi}{2}\), (more of a working condition than a formal one).

This method gave me that \(f\) was maximized when \(\cos(\alpha) = 0, \cos(\beta) = -0.8\) and \(\cos(\gamma) = -0.6\), for which \(f_{\max} = 150\). Plugging these values back into the individual Cosine Law equations gave me the side values noted above. Brian Charlesworth · 2 years, 2 months ago

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@Brian Charlesworth Thanks Sir ! , But Sir Is There any Other Pure Geometric Approach for it ? Thanks Sir Karan Shekhawat · 2 years, 2 months ago

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@Karan Shekhawat I'm pretty sure there is one, but I haven't found it yet. If and when I do find it I will post it here. Brian Charlesworth · 2 years, 2 months ago

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@Deepanshu Gupta Yes My Freind Also Say's that answer is 150 . But Still How did you Solve it by complex number ? :O Karan Shekhawat · 2 years, 2 months ago

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146 when the triangle collapses into a straight line in the limit . That is BC=5+4=9. CA=5 + 3 +8, AB=4-3 =1. So the sum of squares = 81+64+1=146. With say, vertical line CP=5, draw to locii of B and A as circles center P and radii 4 and 3. P should be in the triangle. Collapsed with P would be CPAB. Niranjan Khanderia · 2 years, 2 months ago

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@Niranjan Khanderia I think this would give a maximum value of \(9^{2} + 8^{2} + 1^{2} = 81 + 64 + 1 = 146\), just shy of the maximum of \(150\) found previously. Brian Charlesworth · 2 years, 2 months ago

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@Brian Charlesworth Can you show your method? I have corrected my calculation. Niranjan Khanderia · 2 years, 2 months ago

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@Niranjan Khanderia I've outlined my method above in my response to Deepanshu. His solution, however, is far more elegant than mine. Since the cosine values I found above were so "nice", I suspect that there is a geometric solution as well. Brian Charlesworth · 2 years, 2 months ago

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@Brian Charlesworth Thank you for the method. Niranjan Khanderia · 2 years, 2 months ago

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