relation between inscribed and circumsdribed circle

Question

Let T be the triangle with side lengths $b_1,b_2,b_3$, and $r_{insc}$ and $r_{cir}$ be the radii of the inscribed and circumscribed circles,respectively, I need to show that $$ \frac {r_{insc}}{r_{cir}}=\frac {(b_2+b_3-b_1)(b_3+b_1-b_2)(b_2+b_1-b_3)}{2b_1b_2b_3}$$ Maybe it is easy, but I forgot relations on triangles,

Please help me. Thanks

Answer 1

Let's side of the triangle be $a,b,c$ and $R$ and $r$ are radii of circumscribed and inscribed circles, respectively.

The equaiton now looks like

$$\frac{r}{R} = \frac{(a+b-c)(a+c-b)(b+c-a)}{2abc}$$

We multiply both sides by $\frac{a+b+c}{8}$

$$\frac{r(a+b+c)}{8R} = \frac{a+b-c}{2} \times \frac{a+c-b}{2} \times \frac{b+c-a}{2} \times \frac{a+b+c}{2}\times \frac{1}{abc}$$

$$\frac{r(a+b+c)}{8R} = \frac{(s-c)(s-b)(s-a)s}{abc}$$

Where $s$ is the semiperimetar. By Heron's formula we know that

$$K = \sqrt{(s-c)(s-b)(s-a)s}$$

Where K is the are of the triangle.

$$\frac{r(a+b+c)}{8R} = \frac{K^2}{abc}$$

We substitute $\frac{a+b+c}{2} = s$

$$\frac{rs}{4R} = \frac{K^2}{abc}$$

We know that $K = \frac{abc}{4R} = rs$

$$\frac{rs \times abc}{4R} = K^2$$

$$K^2 = K^2$$

Q.E.D.

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Another solution is making this substitution:

$$R = \frac{abc}{4\sqrt{s(s-a)(s-b)(s-c)}}$$

$$r = \frac{\sqrt{(s-a)(s-b)(s-c)}}{\sqrt{s}}$$

$$\frac{r}{R} = \frac{\frac{\sqrt{(s-a)(s-b)(s-c)}}{\sqrt{s}}}{\frac{abc}{4\sqrt{s(s-a)(s-b)(s-c)}}} = \frac{4(s-a)(s-b)(s-c)}{abc} = \frac{4 \times \frac{c+b-a}{2} \times \frac{a+c-b}{2} \times \frac{a+b-c}{2}}{abc}$$

$$\frac{r}{R} = \frac{(a+b-c)(a+c-b)(c+b-a)}{2abc}$$

Q.E.D.

Answer 2

I use the conventional notation for the proof. Denote $\theta_i$, the angle faced to the side $b_i$ for $i=1,2,3$. It is known that $2R\sin\theta_i=b_i$. Using this you get: $$ b_1+b_2-b_3=8R\sin\frac{\theta_1}{2}\sin\frac{\theta_2}{2}\cos\frac{\theta_3}{2}. $$ You get similar identity for different variation.

Now we have $b_1=r(\cot\frac{\theta_2}{2}+\cot\frac{\theta_3}{2})$ which is: $$ r\cos\frac{\theta_1}{2}=b_1\sin\frac{\theta_3}{2}\sin\frac{\theta_2}{2}. (1) $$ or $$ r\sin{\theta_1}=2b_1\sin\frac{\theta_3}{2}\sin\frac{\theta_2}{2}\sin\frac{\theta_1}{2}. $$ Clearly:

$$ \frac{r}{4R}=\sin\frac{\theta_3}{2}\sin\frac{\theta_2}{2}\sin\frac{\theta_1}{2}. $$ Moreover by multiplying two sides of (1) with two other pissibilities you get: $$ \frac{b_1b_2b_3}{16rR^2}=\cos\frac{\theta_3}{2}\cos\frac{\theta_2}{2}\cos\frac{\theta_1}{2}. $$ Now by simple algebraic manipulation you get the correct result.