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Question Number 214629 by ajfour last updated on 14/Dec/24

Commented by ajfour last updated on 14/Dec/24

$$sayfora=3,b=2.$$

Commented by Ghisom last updated on 24/Dec/24

$$\mathrm{I}\mathrm{get}$$$$r=\frac{ab}{2(a+b)}$$

Commented by ajfour last updated on 24/Dec/24

$$letmetry!$$

Commented by mr W last updated on 24/Dec/24

$${that}^{\prime }scorrect!$$

Answered by ajfour last updated on 24/Dec/24

Commented by ajfour last updated on 25/Dec/24

$$a=\frac{r}{\tan \alpha }+\frac{r}{\tan (\frac{\pi }{4}-\theta )}$$$$say\tan \alpha =t,\tan \theta =m$$$$\Rightarrow \frac{1}{t}=\frac{a}{r}-\left(\frac{1+m}{1-m}\right)or$$$$\frac{m+1}{m-1}=\frac{1}{t}-\frac{a}{r}$$$$\frac{1}{m}=\frac{(\frac{1}{t}-\frac{a}{r})-1}{(\frac{1}{t}-\frac{a}{r})+1}....\left(i\right)$$$$\frac{2t}{1-t^2}=\frac{b}{a}=\frac{1}{k}$$$$t^2+2kt-1=0$$$$t=\sqrt{k^2+1}-k$$$$\frac{r}{\tan \theta }-\frac{r}{\tan (\frac{\pi }{4}+\alpha )}=b$$$$\frac{1}{m}=\frac{b}{r}+\frac{1-t}{1+t}\Rightarrow using..\left(i\right)weget$$$$\frac{(\frac{1}{t}-\frac{a}{r})-1}{(\frac{1}{t}-\frac{a}{r})+1}=\frac{b}{r}+\frac{1-t}{1+t}$$$$eg.a=4,b=3\Rightarrow k=\frac{4}{3},t=\frac{1}{3}$$$$\frac{2-\frac{4}{r}}{4-\frac{4}{r}}=\frac{3}{r}+\frac{1}{2}$$$$\cancel{2}-\frac{4}{r}=\frac{12}{r}+\cancel{2}-\frac{12}{r^2}-\frac{2}{r}$$$$\Rightarrow 14r=12$$$$r=\frac{6}{7}$$

Answered by mr W last updated on 24/Dec/24

Commented by mr W last updated on 24/Dec/24

$$c=\sqrt{a^2+b^2}$$$$AE=AD=\frac{r}{\tan \frac{A}{2}}$$$$CF=CD=a-\frac{r}{\tan \frac{A}{2}}$$$$BK=BG=r\tan \frac{B}{2}$$$$CH=CK=b+r\tan \frac{B}{2}$$$$FH=b+r\tan \frac{B}{2}-(a-\frac{r}{\tan \frac{A}{2}})$$$$GE=c-r\tan \frac{B}{2}-\frac{r}{\tan \frac{A}{2}}$$$$GE=FH$$$$c-r\tan \frac{B}{2}-\frac{r}{\tan \frac{A}{2}}=b+r\tan \frac{B}{2}-a+\frac{r}{\tan \frac{A}{2}}$$$$2r(\tan \frac{B}{2}+\frac{1}{\tan \frac{A}{2}})=c+a-b$$$$2r(\frac{1-\cos B}{\sin B}+\frac{1+\cos A}{\sin A})=c+a-b$$$$2r(\frac{c-b}{a}+\frac{c+a}{b})=c+a-b$$$$2r\left(\frac{cb-b^2+ac+a^2}{ab}\right)=c+a-b$$$$\frac{2r(a+b)(c+a-b)}{ab}=c+a-b$$$$\Rightarrow r=\frac{ab}{2(a+b)}✓$$

Commented by Ghisom last updated on 25/Dec/24

$$\mathrm{yes}.$$

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