Question and Answers Forum

All Questions      Topic List

Geometry Questions

Previous in All Question      Next in All Question      

Previous in Geometry      Next in Geometry      

Question Number 201561 by Mingma last updated on 09/Dec/23

Answered by ajfour last updated on 11/Dec/23

Commented by ajfour last updated on 11/Dec/23

$$2\theta +2\varphi =\frac{\pi }{2}$$$$\tan (\theta +\varphi )=1$$$$\tan \theta +\tan \varphi =1-\tan \theta \tan \varphi$$$$\tan \theta =\frac{q}{r}=\frac{r}{p+a}\Rightarrow pq+aq=r^2$$$$\tan \varphi =\frac{p}{r}=\frac{r}{q+b}\Rightarrow pq+bp=r^2$$$$\Rightarrow \frac{\tan \theta }{\tan \varphi }=\frac{q}{p}\mathrm{\&}aq=bp\left(subtracting\right)$$$$hence\frac{\tan \theta }{\tan \varphi }=\frac{b}{a}$$$$2pq+aq+bp=2r^2\left(adding\right)$$$$2\left(r\tan \varphi \right)\left(r\tan \theta \right)+r(a\tan \theta +b\tan \varphi )$$$$=2r^2$$$$\Rightarrow \frac{1}{2r}(a\tan \theta +b\tan \varphi )=1-\tan \theta \tan \varphi$$$$but1-\tan \theta \tan \varphi =\tan \theta +\tan \varphi$$$$\mathrm{\&}b\tan \varphi =a\tan \theta \Rightarrow$$$$\frac{1}{2r}\left(2a\tan \theta \right)=\tan \theta +\frac{a}{b}\tan \theta$$$$\Rightarrow \frac{1}{r}=\frac{1}{a}+\frac{1}{b}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com