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Modification-of-Q-14157-x-2-y-2-xy-a-2-y-2-z-2-yz-b-2-z-2-x-2-zx-c-2-Pl-discuss-also-geometrical-trigonometrical-aspects-

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Question Number 14364 by RasheedSindhi last updated on 31/May/17
Modification of Q#14157 x^2 +y^2 −xy=a^2 y^2 +z^2 −yz=b^2 z^2 +x^2 −zx=c^2 Pl discuss also geometrical/ trigonometrical aspects.
$$\mathrm{Modification}\:\mathrm{of}\:\mathrm{Q}#\mathrm{14157} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} −\mathrm{xy}=\mathrm{a}^{\mathrm{2}} \\ $$$$\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} −\mathrm{yz}=\mathrm{b}^{\mathrm{2}} \\ $$$$\mathrm{z}^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} −\mathrm{zx}=\mathrm{c}^{\mathrm{2}} \\ $$$$\mathrm{Pl}\:\mathrm{discuss}\:\mathrm{also}\:\mathrm{geometrical}/ \\ $$$$\mathrm{trigonometrical}\:\mathrm{aspects}. \\ $$
Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 31/May/17
mr Rasheed! it is solved.please return to Q.14157.
$${mr}\:{Rasheed}!\:{it}\:{is}\:{solved}.{please}\: \\ $$$${return}\:{to}\:{Q}.\mathrm{14157}. \\ $$
Commented by RasheedSindhi last updated on 31/May/17
Mr/Miss behi.This is somewhat different from Q#14157.Here you can′t draw such triangle as you did in answer of mentioned question.Pl try such geometrical solution here as you did before.
$$\mathrm{Mr}/\mathrm{Miss}\:\mathrm{behi}.\mathrm{This}\:\mathrm{is}\:\mathrm{somewhat} \\ $$$$\mathrm{different}\:\mathrm{from}\:\mathrm{Q}#\mathrm{14157}.\mathrm{Here} \\ $$$$\mathrm{you}\:\mathrm{can}'\mathrm{t}\:\mathrm{draw}\:\mathrm{such}\:\mathrm{triangle} \\ $$$$\mathrm{as}\:\mathrm{you}\:\mathrm{did}\:\mathrm{in}\:\mathrm{answer}\:\mathrm{of}\:\mathrm{mentioned} \\ $$$$\mathrm{question}.\mathrm{Pl}\:\mathrm{try}\:\mathrm{such}\:\mathrm{geometrical} \\ $$$$\mathrm{solution}\:\mathrm{here}\:\mathrm{as}\:\mathrm{you}\:\mathrm{did}\:\mathrm{before}. \\ $$
Commented by mrW1 last updated on 31/May/17
Commented by mrW1 last updated on 31/May/17
This is the geometrical interpretation of the solution.
$${This}\:{is}\:{the}\:{geometrical}\:{interpretation} \\ $$$${of}\:{the}\:{solution}. \\ $$
Commented by RasheedSindhi last updated on 31/May/17
Fine idea! Very general also! It is for x^2 +y^2 +kxy=a^2 for any value of k. My idea is also general but it needs space. See image below At A all the triangular faces have equal angles each equal to 60°.
$$\mathrm{Fine}\:\mathrm{idea}!\:\mathrm{Very}\:\mathrm{general}\:\mathrm{also}!\:\mathrm{It} \\ $$$$\mathrm{is}\:\mathrm{for}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{kxy}=\mathrm{a}^{\mathrm{2}} \:\mathrm{for}\:\mathrm{any} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{k}.\:\mathrm{My}\:\mathrm{idea}\:\mathrm{is}\:\mathrm{also} \\ $$$$\mathrm{general}\:\mathrm{but}\:\mathrm{it}\:\mathrm{needs}\:\mathrm{space}.\: \\ $$$$\mathrm{See}\:\mathrm{image}\:\mathrm{below} \\ $$$$\mathrm{At}\:\mathrm{A}\:\mathrm{all}\:\mathrm{the}\:\mathrm{triangular}\:\mathrm{faces} \\ $$$$\mathrm{have}\:\mathrm{equal}\:\mathrm{angles}\:\mathrm{each}\:\mathrm{equal} \\ $$$$\mathrm{to}\:\mathrm{60}°. \\ $$
Commented by RasheedSindhi last updated on 31/May/17
Commented by mrW1 last updated on 31/May/17
Your idea with the pyramid is remarkable! I thought in plane. But the geometrical meaning from us is the same. If you fold my figure at the lines a,b,c, you will get the same pyramid as yours. Or if you unfold your pyramid, you will get the same figure as mines. I put a more detailed figure.
$${Your}\:{idea}\:{with}\:{the}\:{pyramid}\:{is}\:{remarkable}! \\ $$$$ \\ $$$${I}\:{thought}\:{in}\:{plane}.\:{But}\:{the}\:{geometrical} \\ $$$${meaning}\:{from}\:{us}\:{is}\:{the}\:{same}.\:{If}\:{you} \\ $$$${fold}\:{my}\:{figure}\:{at}\:{the}\:{lines}\:{a},{b},{c},\:{you} \\ $$$${will}\:{get}\:{the}\:{same}\:{pyramid}\:{as}\:{yours}.\:{Or} \\ $$$${if}\:{you}\:{unfold}\:{your}\:{pyramid},\:{you}\:{will} \\ $$$${get}\:{the}\:{same}\:{figure}\:{as}\:{mines}. \\ $$$$ \\ $$$${I}\:{put}\:{a}\:{more}\:{detailed}\:{figure}. \\ $$
Commented by mrW1 last updated on 31/May/17
Commented by RasheedSindhi last updated on 01/Jun/17
Believe me sir, I also discovered same connection after my post! Anyway both are same! The idea came into my mind in this way:Three line segments x,y &z with common endpoint can be drawn at 120° in a plane. Now angles should be decreased to 60°.This can be done by lifting up(or below) non-common endpoints. So in this way the the idea of pyramid came in mind.
$$\mathcal{B}{elieve}\:{me}\:{sir},\:\mathcal{I}\:{also}\:{discovered} \\ $$$${same}\:{connection}\:{after}\:{my}\:{post}! \\ $$$${Anyway}\:{both}\:{are}\:{same}! \\ $$$${The}\:{idea}\:{came}\:{into}\:{my}\:{mind} \\ $$$${in}\:{this}\:{way}:{Three}\:{line}\:{segments} \\ $$$${x},{y}\:\&{z}\:{with}\:{common}\:{endpoint} \\ $$$${can}\:{be}\:{drawn}\:{at}\:\mathrm{120}°\:{in}\:{a}\:{plane}. \\ $$$${Now}\:{angles}\:{should}\:{be}\:{decreased} \\ $$$${to}\:\mathrm{60}°.{This}\:{can}\:{be}\:{done}\:{by}\:{lifting} \\ $$$${up}\left({or}\:{below}\right)\:{non}-{common} \\ $$$${endpoints}.\:{S}\mathrm{o}\:{in}\:{this}\:{way}\:{the} \\ $$$${the}\:{idea}\:{of}\:{pyramid}\:{came}\:{in}\:{mind}. \\ $$

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