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Question-140838

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Question Number 140838 by mr W last updated on 13/May/21
Commented by mr W last updated on 13/May/21
solution to Q140785
$${solution}\:{to}\:{Q}\mathrm{140785} \\ $$
Commented by mr W last updated on 13/May/21
say: O is origin at any position in plane. OA^(→) =A OB^(→) =B OC^(→) =C OD^(→) =D OE^(→) =E (E is any point in plane) ⇒OP^(→) =P=((A+B+C+D)/4) (P is intersection point of bimedians, is also the centroid of quadrilateral) AB^(→) =B−A BC^(→) =C−B CD^(→) =D−C DA^(→) =A−D AE^(→) =E−A PE^(→) =E−P x^2 =∣AE∣^2 =(E−A)∙(E−A)=E^2 +A^2 −2A∙E y^2 =∣BE∣^2 =(E−B)∙(E−B)=E^2 +B^2 −2B∙E z^2 =∣CE∣^2 =(E−C)∙(E−C)=E^2 +C^2 −2C∙E t^2 =∣DE∣^2 =(E−D)∙(E−D)=E^2 +D^2 −2D∙E ∣PE∣^2 =(E−P)∙(E−P)=E^2 +P^2 −2P∙E P^2 =(1/(16))(A+B+C+D)∙(A+B+C+D) P^2 =(1/(16))(A^2 +B^2 +C^2 +D^2 )+(1/8)(A∙B+A∙C+A∙D+B∙C+B∙D+C∙D) x^2 +y^2 +z^2 +t^2 =4E^2 +A^2 +B^2 +C^2 +D^2 −2(A∙E+B∙E+C∙E+D∙E) x^2 +y^2 +z^2 +t^2 =4E^2 +A^2 +B^2 +C^2 +D^2 −2(A+B+C+D)∙E x^2 +y^2 +z^2 +t^2 =4E^2 +A^2 +B^2 +C^2 +D^2 −8P∙E x^2 +y^2 +z^2 +t^2 −4∣PE∣^2 =4E^2 +A^2 +B^2 +C^2 +D^2 −8P∙E−4E^2 −4P^2 +8P∙E x^2 +y^2 +z^2 +t^2 −4∣PE∣^2 =A^2 +B^2 +C^2 +D^2 −4P^2 x^2 +y^2 +z^2 +t^2 −4∣PE∣^2 =A^2 +B^2 +C^2 +D^2 −(1/4)(A^2 +B^2 +C^2 +D^2 )−(1/2)(A∙B+A∙C+A∙D+B∙C+B∙D+C∙D) x^2 +y^2 +z^2 +t^2 −4∣PE∣^2 =(3/4)(A^2 +B^2 +C^2 +D^2 )−(1/2)(A∙B+A∙C+A∙D+B∙C+B∙D+C∙D) a^2 =∣AB∣^2 =(B−A)∙(B−A)=B^2 +A^2 −2A∙B b^2 =∣BC∣^2 =(C−B)∙(C−B)=C^2 +B^2 −2B∙C c^2 =∣CD∣^2 =(D−C)∙(D−C)=D^2 +C^2 −2C∙D d^2 =∣DA∣^2 =(A−D)∙(A−D)=A^2 +D^2 −2A∙D e^2 =∣AC∣^2 =(C−A)∙(C−A)=C^2 +A^2 −2A∙C f^2 =∣BD∣^2 =(D−B)∙(D−B)=D^2 +B^2 −2B∙D a^2 +b^2 +c^2 +d^2 +e^2 +f^2 =3(A^2 +B^2 +C^2 +D^2 )−2(A∙B+A∙C+A∙D+B∙C+B∙D+C∙D) (1/4)(a^2 +b^2 +c^2 +d^2 +e^2 +f^2 )=(3/4)(A^2 +B^2 +C^2 +D^2 )−(1/2)(A∙B+A∙C+A∙D+B∙C+B∙D+C∙D) ⇒(1/4)(a^2 +b^2 +c^2 +d^2 +e^2 +f^2 )=x^2 +y^2 +z^2 +t^2 −4∣PE∣^2 ⇒(1/4)(a^2 +b^2 +c^2 +d^2 +e^2 +f^2 )+4∣PE∣^2 =x^2 +y^2 +z^2 +t^2
$${say}: \\ $$$${O}\:{is}\:{origin}\:{at}\:{any}\:{position}\:{in}\:{plane}. \\ $$$$\overset{\rightarrow} {{OA}}=\boldsymbol{{A}} \\ $$$$\overset{\rightarrow} {{OB}}=\boldsymbol{{B}} \\ $$$$\overset{\rightarrow} {{OC}}=\boldsymbol{{C}} \\ $$$$\overset{\rightarrow} {{OD}}=\boldsymbol{{D}} \\ $$$$\overset{\rightarrow} {{OE}}=\boldsymbol{{E}}\:\:\left({E}\:{is}\:{any}\:{point}\:{in}\:{plane}\right) \\ $$$$\Rightarrow\overset{\rightarrow} {{OP}}=\boldsymbol{{P}}=\frac{\boldsymbol{{A}}+\boldsymbol{{B}}+\boldsymbol{{C}}+\boldsymbol{{D}}}{\mathrm{4}} \\ $$$$\left({P}\:{is}\:{intersection}\:{point}\:{of}\:{bimedians},\right. \\ $$$$\left.{is}\:{also}\:{the}\:{centroid}\:{of}\:{quadrilateral}\right) \\ $$$$ \\ $$$$\overset{\rightarrow} {{AB}}=\boldsymbol{{B}}−\boldsymbol{{A}} \\ $$$$\overset{\rightarrow} {{BC}}=\boldsymbol{{C}}−\boldsymbol{{B}} \\ $$$$\overset{\rightarrow} {{CD}}=\boldsymbol{{D}}−\boldsymbol{{C}} \\ $$$$\overset{\rightarrow} {{DA}}=\boldsymbol{{A}}−\boldsymbol{{D}} \\ $$$$\overset{\rightarrow} {{AE}}=\boldsymbol{{E}}−\boldsymbol{{A}} \\ $$$$\overset{\rightarrow} {{PE}}=\boldsymbol{{E}}−\boldsymbol{{P}} \\ $$$$ \\ $$$${x}^{\mathrm{2}} =\mid\boldsymbol{{AE}}\mid^{\mathrm{2}} =\left(\boldsymbol{{E}}−\boldsymbol{{A}}\right)\centerdot\left(\boldsymbol{{E}}−\boldsymbol{{A}}\right)=\boldsymbol{{E}}^{\mathrm{2}} +\boldsymbol{{A}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{A}}\centerdot\boldsymbol{{E}} \\ $$$${y}^{\mathrm{2}} =\mid\boldsymbol{{BE}}\mid^{\mathrm{2}} =\left(\boldsymbol{{E}}−\boldsymbol{{B}}\right)\centerdot\left(\boldsymbol{{E}}−\boldsymbol{{B}}\right)=\boldsymbol{{E}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{B}}\centerdot\boldsymbol{{E}} \\ $$$${z}^{\mathrm{2}} =\mid\boldsymbol{{CE}}\mid^{\mathrm{2}} =\left(\boldsymbol{{E}}−\boldsymbol{{C}}\right)\centerdot\left(\boldsymbol{{E}}−\boldsymbol{{C}}\right)=\boldsymbol{{E}}^{\mathrm{2}} +\boldsymbol{{C}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{C}}\centerdot\boldsymbol{{E}} \\ $$$${t}^{\mathrm{2}} =\mid\boldsymbol{{DE}}\mid^{\mathrm{2}} =\left(\boldsymbol{{E}}−\boldsymbol{{D}}\right)\centerdot\left(\boldsymbol{{E}}−\boldsymbol{{D}}\right)=\boldsymbol{{E}}^{\mathrm{2}} +\boldsymbol{{D}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{D}}\centerdot\boldsymbol{{E}} \\ $$$$\mid\boldsymbol{{PE}}\mid^{\mathrm{2}} =\left(\boldsymbol{{E}}−\boldsymbol{{P}}\right)\centerdot\left(\boldsymbol{{E}}−\boldsymbol{{P}}\right)=\boldsymbol{{E}}^{\mathrm{2}} +\boldsymbol{{P}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{P}}\centerdot\boldsymbol{{E}} \\ $$$$ \\ $$$$\boldsymbol{{P}}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{16}}\left(\boldsymbol{{A}}+\boldsymbol{{B}}+\boldsymbol{{C}}+\boldsymbol{{D}}\right)\centerdot\left(\boldsymbol{{A}}+\boldsymbol{{B}}+\boldsymbol{{C}}+\boldsymbol{{D}}\right) \\ $$$$\boldsymbol{{P}}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{16}}\left(\boldsymbol{{A}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} +\boldsymbol{{C}}^{\mathrm{2}} +\boldsymbol{{D}}^{\mathrm{2}} \right)+\frac{\mathrm{1}}{\mathrm{8}}\left(\boldsymbol{{A}}\centerdot\boldsymbol{{B}}+\boldsymbol{{A}}\centerdot\boldsymbol{{C}}+\boldsymbol{{A}}\centerdot\boldsymbol{{D}}+\boldsymbol{{B}}\centerdot\boldsymbol{{C}}+\boldsymbol{{B}}\centerdot\boldsymbol{{D}}+\boldsymbol{{C}}\centerdot\boldsymbol{{D}}\right) \\ $$$$ \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +{t}^{\mathrm{2}} =\mathrm{4}\boldsymbol{{E}}^{\mathrm{2}} +\boldsymbol{{A}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} +\boldsymbol{{C}}^{\mathrm{2}} +\boldsymbol{{D}}^{\mathrm{2}} −\mathrm{2}\left(\boldsymbol{{A}}\centerdot\boldsymbol{{E}}+\boldsymbol{{B}}\centerdot\boldsymbol{{E}}+\boldsymbol{{C}}\centerdot\boldsymbol{{E}}+\boldsymbol{{D}}\centerdot\boldsymbol{{E}}\right) \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +{t}^{\mathrm{2}} =\mathrm{4}\boldsymbol{{E}}^{\mathrm{2}} +\boldsymbol{{A}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} +\boldsymbol{{C}}^{\mathrm{2}} +\boldsymbol{{D}}^{\mathrm{2}} −\mathrm{2}\left(\boldsymbol{{A}}+\boldsymbol{{B}}+\boldsymbol{{C}}+\boldsymbol{{D}}\right)\centerdot\boldsymbol{{E}} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +{t}^{\mathrm{2}} =\mathrm{4}\boldsymbol{{E}}^{\mathrm{2}} +\boldsymbol{{A}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} +\boldsymbol{{C}}^{\mathrm{2}} +\boldsymbol{{D}}^{\mathrm{2}} −\mathrm{8}\boldsymbol{{P}}\centerdot\boldsymbol{{E}} \\ $$$$ \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +{t}^{\mathrm{2}} −\mathrm{4}\mid\boldsymbol{{PE}}\mid^{\mathrm{2}} =\mathrm{4}\boldsymbol{{E}}^{\mathrm{2}} +\boldsymbol{{A}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} +\boldsymbol{{C}}^{\mathrm{2}} +\boldsymbol{{D}}^{\mathrm{2}} −\mathrm{8}\boldsymbol{{P}}\centerdot\boldsymbol{{E}}−\mathrm{4}\boldsymbol{{E}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{{P}}^{\mathrm{2}} +\mathrm{8}\boldsymbol{{P}}\centerdot\boldsymbol{{E}} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +{t}^{\mathrm{2}} −\mathrm{4}\mid\boldsymbol{{PE}}\mid^{\mathrm{2}} =\boldsymbol{{A}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} +\boldsymbol{{C}}^{\mathrm{2}} +\boldsymbol{{D}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{{P}}^{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +{t}^{\mathrm{2}} −\mathrm{4}\mid\boldsymbol{{PE}}\mid^{\mathrm{2}} =\boldsymbol{{A}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} +\boldsymbol{{C}}^{\mathrm{2}} +\boldsymbol{{D}}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{4}}\left(\boldsymbol{{A}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} +\boldsymbol{{C}}^{\mathrm{2}} +\boldsymbol{{D}}^{\mathrm{2}} \right)−\frac{\mathrm{1}}{\mathrm{2}}\left(\boldsymbol{{A}}\centerdot\boldsymbol{{B}}+\boldsymbol{{A}}\centerdot\boldsymbol{{C}}+\boldsymbol{{A}}\centerdot\boldsymbol{{D}}+\boldsymbol{{B}}\centerdot\boldsymbol{{C}}+\boldsymbol{{B}}\centerdot\boldsymbol{{D}}+\boldsymbol{{C}}\centerdot\boldsymbol{{D}}\right) \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +{t}^{\mathrm{2}} −\mathrm{4}\mid\boldsymbol{{PE}}\mid^{\mathrm{2}} =\frac{\mathrm{3}}{\mathrm{4}}\left(\boldsymbol{{A}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} +\boldsymbol{{C}}^{\mathrm{2}} +\boldsymbol{{D}}^{\mathrm{2}} \right)−\frac{\mathrm{1}}{\mathrm{2}}\left(\boldsymbol{{A}}\centerdot\boldsymbol{{B}}+\boldsymbol{{A}}\centerdot\boldsymbol{{C}}+\boldsymbol{{A}}\centerdot\boldsymbol{{D}}+\boldsymbol{{B}}\centerdot\boldsymbol{{C}}+\boldsymbol{{B}}\centerdot\boldsymbol{{D}}+\boldsymbol{{C}}\centerdot\boldsymbol{{D}}\right) \\ $$$$ \\ $$$${a}^{\mathrm{2}} =\mid\boldsymbol{{AB}}\mid^{\mathrm{2}} =\left(\boldsymbol{{B}}−\boldsymbol{{A}}\right)\centerdot\left(\boldsymbol{{B}}−\boldsymbol{{A}}\right)=\boldsymbol{{B}}^{\mathrm{2}} +\boldsymbol{{A}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{A}}\centerdot\boldsymbol{{B}} \\ $$$${b}^{\mathrm{2}} =\mid\boldsymbol{{BC}}\mid^{\mathrm{2}} =\left(\boldsymbol{{C}}−\boldsymbol{{B}}\right)\centerdot\left(\boldsymbol{{C}}−\boldsymbol{{B}}\right)=\boldsymbol{{C}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{B}}\centerdot\boldsymbol{{C}} \\ $$$${c}^{\mathrm{2}} =\mid\boldsymbol{{CD}}\mid^{\mathrm{2}} =\left(\boldsymbol{{D}}−\boldsymbol{{C}}\right)\centerdot\left(\boldsymbol{{D}}−\boldsymbol{{C}}\right)=\boldsymbol{{D}}^{\mathrm{2}} +\boldsymbol{{C}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{C}}\centerdot\boldsymbol{{D}} \\ $$$${d}^{\mathrm{2}} =\mid\boldsymbol{{DA}}\mid^{\mathrm{2}} =\left(\boldsymbol{{A}}−\boldsymbol{{D}}\right)\centerdot\left(\boldsymbol{{A}}−\boldsymbol{{D}}\right)=\boldsymbol{{A}}^{\mathrm{2}} +\boldsymbol{{D}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{A}}\centerdot\boldsymbol{{D}} \\ $$$${e}^{\mathrm{2}} =\mid\boldsymbol{{AC}}\mid^{\mathrm{2}} =\left(\boldsymbol{{C}}−\boldsymbol{{A}}\right)\centerdot\left(\boldsymbol{{C}}−\boldsymbol{{A}}\right)=\boldsymbol{{C}}^{\mathrm{2}} +\boldsymbol{{A}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{A}}\centerdot\boldsymbol{{C}} \\ $$$${f}^{\mathrm{2}} =\mid\boldsymbol{{BD}}\mid^{\mathrm{2}} =\left(\boldsymbol{{D}}−\boldsymbol{{B}}\right)\centerdot\left(\boldsymbol{{D}}−\boldsymbol{{B}}\right)=\boldsymbol{{D}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} −\mathrm{2}\boldsymbol{{B}}\centerdot\boldsymbol{{D}} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} +{e}^{\mathrm{2}} +{f}^{\mathrm{2}} =\mathrm{3}\left(\boldsymbol{{A}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} +\boldsymbol{{C}}^{\mathrm{2}} +\boldsymbol{{D}}^{\mathrm{2}} \right)−\mathrm{2}\left(\boldsymbol{{A}}\centerdot\boldsymbol{{B}}+\boldsymbol{{A}}\centerdot\boldsymbol{{C}}+\boldsymbol{{A}}\centerdot\boldsymbol{{D}}+\boldsymbol{{B}}\centerdot\boldsymbol{{C}}+\boldsymbol{{B}}\centerdot\boldsymbol{{D}}+\boldsymbol{{C}}\centerdot\boldsymbol{{D}}\right) \\ $$$$\frac{\mathrm{1}}{\mathrm{4}}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} +{e}^{\mathrm{2}} +{f}^{\mathrm{2}} \right)=\frac{\mathrm{3}}{\mathrm{4}}\left(\boldsymbol{{A}}^{\mathrm{2}} +\boldsymbol{{B}}^{\mathrm{2}} +\boldsymbol{{C}}^{\mathrm{2}} +\boldsymbol{{D}}^{\mathrm{2}} \right)−\frac{\mathrm{1}}{\mathrm{2}}\left(\boldsymbol{{A}}\centerdot\boldsymbol{{B}}+\boldsymbol{{A}}\centerdot\boldsymbol{{C}}+\boldsymbol{{A}}\centerdot\boldsymbol{{D}}+\boldsymbol{{B}}\centerdot\boldsymbol{{C}}+\boldsymbol{{B}}\centerdot\boldsymbol{{D}}+\boldsymbol{{C}}\centerdot\boldsymbol{{D}}\right) \\ $$$$ \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{4}}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} +{e}^{\mathrm{2}} +{f}^{\mathrm{2}} \right)={x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +{t}^{\mathrm{2}} −\mathrm{4}\mid\boldsymbol{{PE}}\mid^{\mathrm{2}} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{4}}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} +{e}^{\mathrm{2}} +{f}^{\mathrm{2}} \right)+\mathrm{4}\mid\boldsymbol{{PE}}\mid^{\mathrm{2}} ={x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} +{t}^{\mathrm{2}} \\ $$
Commented by mathdanisur last updated on 13/May/21
thank you dear Sir cool
$${thank}\:{you}\:{dear}\:{Sir}\:{cool} \\ $$
Commented by BHOOPENDRA last updated on 13/May/21
Nice solution sir
$${Nice}\:{solution}\:{sir} \\ $$

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