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Question Number 155770 by amin96 last updated on 04/Oct/21

Commented by amin96 last updated on 04/Oct/21

$$\mathbf{P}rovethat2\mathbf{mn}=\mathbf{D}\times \mathbf{d}$$

Answered by mr W last updated on 05/Oct/21

Commented by mr W last updated on 04/Oct/21

$$sayradiusofbigcircleisR.$$$${OA}^2=R^2-\frac{D^2}{4}$$$${OB}^2=R^2-\frac{d^2}{4}$$$$CD=\frac{m-n}{2}$$$${OD}^2=R^2-{\left(\frac{m+n}{2}\right)}^2$$$${OB}^2={(OD+CB)}^2+{CD}^2$$$${OB}^2={OD}^2+{CB}^2+2\times OD\times CB+{CD}^2$$$$R^2-\frac{d^2}{4}=R^2-{\left(\frac{m+n}{2}\right)}^2+{\left(\frac{d}{2}\right)}^2+d\sqrt{R^2-{\left(\frac{m+n}{2}\right)}^2}+{\left(\frac{m-n}{2}\right)}^2$$$$\Rightarrow 2mn=d^2+2d\sqrt{R^2-{\left(\frac{m+n}{2}\right)}^2}...\left(i\right)$$$${OA}^2={(CA-OD)}^2+{CD}^2$$$${OA}^2={CA}^2+{OD}^2-2\times OD\times CA+{CD}^2$$$$R^2-\frac{D^2}{4}={\left(\frac{D}{2}\right)}^2+R^2-{\left(\frac{m+n}{2}\right)}^2-D\sqrt{R^2-{\left(\frac{m+n}{2}\right)}^2}+{\left(\frac{m-n}{2}\right)}^2$$$$\Rightarrow 2mn=D^2-2D\sqrt{R^2-{\left(\frac{m+n}{2}\right)}^2}...\left(ii\right)$$$$\left(i\right)\times D+\left(ii\right)\times d:$$$$2(D+d)mn=d^{2}D+D^{2}d=(D+d)Dd$$$$\Rightarrow 2mn=Dd$$

Commented by mr W last updated on 04/Oct/21

$$specialcasewithD=d=x:$$$$Q152486$$

Commented by mr W last updated on 04/Oct/21

Commented by Tawa11 last updated on 04/Oct/21

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

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