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Question Number 185851 by Mingma last updated on 28/Jan/23

Commented by Mingma last updated on 28/Jan/23

Diameter in relation to a,b,c ?

Answered by mr W last updated on 28/Jan/23

Commented by mr W last updated on 28/Jan/23

$$R^2={(c-R)}^2+{\left(\frac{a+b}{2}\right)}^2$$$$\Rightarrow R=\frac{c}{2}+\frac{{(a+b)}^2}{8c}$$$$OC=R-r$$$$OB=c-R+r$$$$BC=\frac{a+b}{2}-a=\frac{b-a}{2}$$$${(R-r)}^2={(c-R+r)}^2+{\left(\frac{b-a}{2}\right)}^2$$$$\Rightarrow r=R-\frac{c}{2}-\frac{{(b-a)}^2}{8c}$$$$\Rightarrow r=\frac{{(a+b)}^2}{8c}-\frac{{(b-a)}^2}{8c}=\frac{ab}{2c}$$$$\Rightarrow d=2r=\frac{ab}{c}✓$$$$\Rightarrow d=\frac{3\times 6}{9}=2$$

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