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# In triangle , , , and . Let be the incenter. The incircle of triangle touches sides , , and at , , and , respectively. Find the length of .

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In triangle A, B, C, AB=13, AC=15 and BC=14. Let I be the incenter. The incircle of triangle ABC touches sides BC, AC, and AB at D, E, and F, respectively. Find the length of BI.

Aug 9, 2018

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In triangle A, B, C, AB=13, AC=15 and BC=14. Let I be the incenter. The incircle of triangle ABC touches sides BC, AC, and AB at D, E, and F, respectively. Find the length of BI.

Hello friends!

triangle ABC

a=14   b=15   c=13

Referring to the graph in Mathhemathh's reply on August 09, 2018.

What is the radius of the circle inscribed in triangle ABC if AB = 5, AC=6, BC=7?

formula of Heron

$$A=\sqrt{s(s-a)(s-b)(s-c)}\\ s=\frac{1}{2}(a+b+c)\\ s=\frac{1}{2}(14+15+13)\\ s=21\\ A=\sqrt{21(21-14)(21-15)(21-13)}$$

$$A=84$$

$$A=\frac{ar+br+cr}{2}\\ r=\frac{2A}{a+b+c}=\frac{168}{14+15+13}=\frac{168}{42}\\ \color{blue}r=4$$

Thanks Mathhemathh!

2. angle $$\beta$$

cosine

$$b^2=a^2+c^2-2ac\cdot cos \beta\\ \beta=arccos\frac{a^2+c^2-b^2}{2ac}\\ \beta=arccos\frac{14^2+13^2-15^2}{2\cdot 14\cdot 13c}=arccos\ 0.384615$$

$$\beta=67.3801°$$

$$\frac{\beta}{2}=33.690°$$

3. Length of the route $$\overline{BI}$$

$$sin\frac{\beta}{2}=\frac{r}{\overline{BI}}\\ \overline{BI}=\frac{r}{sin\frac{\beta}{2}}=\frac{4}{sin\ 33.690°}$$

$$\overline{BI}=7.2111$$

$$The\ length\ of\ \overline{BI}\ is\ 7,2111$$ !

Aug 11, 2018
edited by asinus  Aug 11, 2018
edited by asinus  Aug 11, 2018
edited by asinus  Aug 11, 2018
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In triangle A, B, C, AB=13, AC=15 and BC=14. Let I be the incenter. The incircle of triangle ABC touches sides BC, AC, and AB at D, E, and F, respectively. Find the length of BI.   Aug 11, 2018