Menu Close

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-




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.
$$\mathcal{B}{elieve}\:{me}\:{sir},\:\mathcal{I}\:{also}\:{discovered} \\ $$$${same}\:{connection}\:{after}\:{my}\:{post}! \\ $$$${Anyway}\:{both}\:{are}\:{same}! \\ $$$${The}\:{idea}\:{came}\:{into}\:{my}\:{mind} \\ $$$${in}\:{this}\:{way}:{Three}\:{line}\:{segments} \\ $$$${x},{y}\:\&{z}\:{with}\:{common}\:{endpoint} \\ $$$${can}\:{be}\:{drawn}\:{at}\:\mathrm{120}°\:{in}\:{a}\:{plane}. \\ $$$${Now}\:{angles}\:{should}\:{be}\:{decreased} \\ $$$${to}\:\mathrm{60}°.{This}\:{can}\:{be}\:{done}\:{by}\:{lifting} \\ $$$${up}\left({or}\:{below}\right)\:{non}-{common} \\ $$$${endpoints}.\:{S}\mathrm{o}\:{in}\:{this}\:{way}\:{the} \\ $$$${the}\:{idea}\:{of}\:{pyramid}\:{came}\:{in}\:{mind}. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *