Question Number 222197 by BHOOPENDRA last updated on 20/Jun/25
$${question}\:\mathrm{211277} \\ $$
Answered by BHOOPENDRA last updated on 20/Jun/25
Commented by BHOOPENDRA last updated on 23/Jun/25
Commented by BHOOPENDRA last updated on 23/Jun/25
Commented by mr W last updated on 21/Jun/25
$${can}\:{you}\:{determine}\:\:{d},\:{e},\:{f}\:{generally} \\ $$$${in}\:{terms}\:{of}\:{a},\:{b},\:{c},\:\theta?\: \\ $$$${or}\:{at}\:{least}\:{for}\:\theta=\mathrm{60}°. \\ $$
Commented by mr W last updated on 21/Jun/25
$${great}! \\ $$$${till}\:{now}\:{we}\:{only}\:{have}\:{exact}\:{solution} \\ $$$${for}\:{the}\:{cases}\:\lambda=\mathrm{1}\:\left(\theta=\mathrm{120}°\right)\:{and}\: \\ $$$$\lambda=\mathrm{0}\:\left(\theta=\mathrm{90}°\right). \\ $$
Commented by mr W last updated on 20/Jun/25
![the task is to solve following real equation system: y^2 +z^2 +λyz=a^2 z^2 +x^2 +λzx=b^2 x^2 +y^2 +λxy=c^2 with λ=−2 cos θ ∈(−2, 1]](https://www.tinkutara.com/question/Q222215.png)
$${the}\:{task}\:{is}\:{to}\:{solve}\:{following}\:{real} \\ $$$${equation}\:{system}: \\ $$$${y}^{\mathrm{2}} +{z}^{\mathrm{2}} +\lambda{yz}={a}^{\mathrm{2}} \\ $$$${z}^{\mathrm{2}} +{x}^{\mathrm{2}} +\lambda{zx}={b}^{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\lambda{xy}={c}^{\mathrm{2}} \\ $$$${with}\:\lambda=−\mathrm{2}\:\mathrm{cos}\:\theta\:\in\left(−\mathrm{2},\:\mathrm{1}\right] \\ $$
Commented by BHOOPENDRA last updated on 20/Jun/25
$${yes}\:{actully}\:{i}\:{have}\:{tried}\:{to}\:{send}\:{that}\:{solution}\:{so}\:{many}\: \\ $$$${times}\:{i}\:{dont}\:{know}\:{why}\:{its}\:{showing}\: \\ $$$${try}\:{again}\:{give}\:{me}\:{some}\:{time}\:{i}\:{ll}\:{try}\: \\ $$$${to}\:{send}\:{another}\:{way} \\ $$$$ \\ $$
Commented by BHOOPENDRA last updated on 20/Jun/25
Commented by BHOOPENDRA last updated on 20/Jun/25
Commented by BHOOPENDRA last updated on 20/Jun/25
Commented by BHOOPENDRA last updated on 20/Jun/25
Commented by BHOOPENDRA last updated on 20/Jun/25
Commented by mr W last updated on 20/Jun/25
$${thanks}\:{sir}! \\ $$$${the}\:{formula}\:{for}\:{volume}\:{of}\:{a} \\ $$$${tetrahedron}\:{in}\:{terms}\:{of}\:{its}\:\mathrm{6}\:{edge}\: \\ $$$${lengthes}\:{is}\:{known},\:{see}\:{also} \\ $$$${Q}\mathrm{40469}. \\ $$$${so}\:{the}\:{question}\:{here}\:{is}\:{how}\:{to}\:{find} \\ $$$${d},\:{e},\:{f}\:{in}\:{terms}\:{of}\:{a},{b},{c},\theta.\:\:{sorry},\:{but} \\ $$$${i}\:{can}'{t}\:{see}\:{in}\:{your}\:{solution}\:{how}\:{to} \\ $$$${do}\:{this}.\:{please}\:{enlighten}\:{me}. \\ $$
Commented by BHOOPENDRA last updated on 21/Jun/25
Commented by BHOOPENDRA last updated on 21/Jun/25
$${again}\:{the}\:{same}\:{problem}\:{Sir}\:{i}\:{tried}\:{to}\:{attach} \\ $$$${that}\:{image}\:{its}\:{showing}\:{error}\:{but}\:{the}\: \\ $$$${result}\:{i}\:{am}\:{attaching}\:{once}\:{i}\:{ll}\:{be}\: \\ $$$${i}\:{ll}\:{try}\:{to}\:{type}\:{on}\:{this}\:{app}\:. \\ $$
Commented by BHOOPENDRA last updated on 21/Jun/25
$${It}\:{seems}\:{that}\:{sir}\:{if}\:{a},{b},{c}\:{form}\:{a}\: \\ $$$${right}\:{angle}\:{triangle},{there}\:{are}\:{no}\:{solution} \\ $$$${The}\:\mathrm{13},\mathrm{14},\mathrm{15}\:{base}\:{triangle}\:{has}\:\mathrm{2}\: \\ $$$${solutions} \\ $$
Commented by mr W last updated on 21/Jun/25
$${i}\:{can}'{t}\:{see}\:{a}\:{reason}\:{why}\:{there}\:{is}\:{no} \\ $$$${solution}\:{if}\:{a},\:{b},\:{c}\:{form}\:{a}\:{right}\:{angled} \\ $$$${triangle}.\:{but}\:{i}\:{can}\:{image}\:{that}\:{there} \\ $$$${are}\:{more}\:{than}\:{one}\:{solution}\:{in}\:{some} \\ $$$${cases}. \\ $$
Commented by BHOOPENDRA last updated on 21/Jun/25
Commented by BHOOPENDRA last updated on 21/Jun/25
$${In}\:{some}\:{cases}\:{there}\:{is}\:{no}\:{solution}\: \\ $$$${because}\:{what}\:{if}\:{e}>{a} \\ $$
Commented by BHOOPENDRA last updated on 23/Jun/25
Commented by BHOOPENDRA last updated on 23/Jun/25
Commented by Nicholas666 last updated on 23/Jun/25
$$\boldsymbol{\mathrm{amazing}}\centerdot\centerdot\centerdot \\ $$