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Question-168550

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Question Number 168550 by Shirnivak last updated on 13/Apr/22
Answered by mr W last updated on 13/Apr/22
Commented by mr W last updated on 13/Apr/22
R=radius of circumcircle of ΔABC OA=OB=(c/( (√2))) AG=AC=b BD=BC=a AF=(√2)b BE=(√2)a OG^2 =((c/( (√2))))^2 +b^2 −2×(c/( (√2)))×b cos (A+(π/2)+(π/4)) OG^2 =(c^2 /( 2))+b^2 +(√2)cb sin (A+(π/4)) OG^2 =(c^2 /( 2))+b^2 +cb (sin A+cos A) OG^2 =(c^2 /( 2))+b^2 +cb ((a/(2R))+((c^2 +b^2 −a^2 )/(2cb))) OG^2 =c^2 +((abc)/(2R))+((3b^2 −a^2 )/2) OE^2 =((c/( (√2))))^2 +((√2)a)^2 −2ca cos (B+(π/4)+(π/4)) OE^2 =(c^2 /( 2))+2a^2 +2ca sin B OE^2 =(c^2 /( 2))+2a^2 +((abc)/R) OG^2 +OE^2 =(3/2)(a^2 +b^2 +c^2 +((abc)/R)) similarly OD^2 +OF^2 =(3/2)(a^2 +b^2 +c^2 +((abc)/R)) ⇒OG^2 +OE^2 =OD^2 +OF^2
$${R}={radius}\:{of}\:{circumcircle}\:{of}\:\Delta{ABC} \\ $$$${OA}={OB}=\frac{{c}}{\:\sqrt{\mathrm{2}}} \\ $$$${AG}={AC}={b} \\ $$$${BD}={BC}={a} \\ $$$${AF}=\sqrt{\mathrm{2}}{b} \\ $$$${BE}=\sqrt{\mathrm{2}}{a} \\ $$$${OG}^{\mathrm{2}} =\left(\frac{{c}}{\:\sqrt{\mathrm{2}}}\right)^{\mathrm{2}} +{b}^{\mathrm{2}} −\mathrm{2}×\frac{{c}}{\:\sqrt{\mathrm{2}}}×{b}\:\mathrm{cos}\:\left({A}+\frac{\pi}{\mathrm{2}}+\frac{\pi}{\mathrm{4}}\right) \\ $$$${OG}^{\mathrm{2}} =\frac{{c}^{\mathrm{2}} }{\:\mathrm{2}}+{b}^{\mathrm{2}} +\sqrt{\mathrm{2}}{cb}\:\mathrm{sin}\:\left({A}+\frac{\pi}{\mathrm{4}}\right) \\ $$$${OG}^{\mathrm{2}} =\frac{{c}^{\mathrm{2}} }{\:\mathrm{2}}+{b}^{\mathrm{2}} +{cb}\:\left(\mathrm{sin}\:{A}+\mathrm{cos}\:{A}\right) \\ $$$${OG}^{\mathrm{2}} =\frac{{c}^{\mathrm{2}} }{\:\mathrm{2}}+{b}^{\mathrm{2}} +{cb}\:\left(\frac{{a}}{\mathrm{2}{R}}+\frac{{c}^{\mathrm{2}} +{b}^{\mathrm{2}} −{a}^{\mathrm{2}} }{\mathrm{2}{cb}}\right) \\ $$$${OG}^{\mathrm{2}} ={c}^{\mathrm{2}} +\frac{{abc}}{\mathrm{2}{R}}+\frac{\mathrm{3}{b}^{\mathrm{2}} −{a}^{\mathrm{2}} }{\mathrm{2}} \\ $$$${OE}^{\mathrm{2}} =\left(\frac{{c}}{\:\sqrt{\mathrm{2}}}\right)^{\mathrm{2}} +\left(\sqrt{\mathrm{2}}{a}\right)^{\mathrm{2}} −\mathrm{2}{ca}\:\mathrm{cos}\:\left({B}+\frac{\pi}{\mathrm{4}}+\frac{\pi}{\mathrm{4}}\right) \\ $$$${OE}^{\mathrm{2}} =\frac{{c}^{\mathrm{2}} }{\:\mathrm{2}}+\mathrm{2}{a}^{\mathrm{2}} +\mathrm{2}{ca}\:\mathrm{sin}\:{B} \\ $$$${OE}^{\mathrm{2}} =\frac{{c}^{\mathrm{2}} }{\:\mathrm{2}}+\mathrm{2}{a}^{\mathrm{2}} +\frac{{abc}}{{R}} \\ $$$${OG}^{\mathrm{2}} +{OE}^{\mathrm{2}} =\frac{\mathrm{3}}{\mathrm{2}}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +\frac{{abc}}{{R}}\right) \\ $$$${similarly} \\ $$$${OD}^{\mathrm{2}} +{OF}^{\mathrm{2}} =\frac{\mathrm{3}}{\mathrm{2}}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +\frac{{abc}}{{R}}\right) \\ $$$$\Rightarrow{OG}^{\mathrm{2}} +{OE}^{\mathrm{2}} ={OD}^{\mathrm{2}} +{OF}^{\mathrm{2}} \\ $$
Commented by Tawa11 last updated on 13/Apr/22
Great sir.
$$\mathrm{Great}\:\mathrm{sir}. \\ $$

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