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ABC is a right triangle at $A$ , $BH , CD$ are the bisectors of angles $\angle {B},\angle { C}$, respectively , if $BH=9$ and $CD=8\sqrt {2}$ , Find the length of $BC$?

user373141
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    Please see the faq section for tips on asking a good question. This style of posting (dumping a problem statement) is not acceptable according to our guidelines. You should work to improve it pronto. – rschwieb Dec 22 '17 at 21:35
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    Draw a diagram and use some theorems. That's how geometry works – Yuriy S Dec 22 '17 at 21:41
  • @user373141 If you are ok, you can accept the answer and set as solved. Thanks! – user Dec 23 '17 at 18:44

2 Answers2

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HINT

Let $a,b,c$ the side lengths BC, AC, AB and $2p=a+b+c$, then:

$$BH=\frac{2}{a+c}\sqrt{acp(p-b)}$$

$$CD=\frac{2}{a+b}\sqrt{abp(p-c)}$$

user
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or use that $$\sqrt{\frac{s(s-b)}{ac}}=\frac{c}{9},\sqrt{\frac{s(s-c)}{ba}}=\frac{b}{8\sqrt{2}},\frac{a+b+c}{2}=s,a=\sqrt{c^2+b^2}$$

Dr. Sonnhard Graubner
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