# The Extended Law of Sines

*Well, I start the collection with one of the most importand theorems in Geometry, the sines law for every triangle. I think this blog must start with this basic theorem because many others are proved usind this.
*

**Theorem**

In any triangle with angles and sides respectively the following is true

where is the circumradii of the triangle.

**Proofs**

**Proof 1 **

Let be an acute triangle and be its circumcircle. Suppose the diameter of and bring the chord . Then

and in the right triangle ,

So

** **

If was was an obtuse one, we would follow the same way with the difference that and still

If, finally, was a right triangle, then the proof would be obvious. *(I leave this proof for the reader :))*

*Proof 2 **(Using Area formulas)*

The area of every triangle is given by and (see the topic about Area’s Formulas which i have not yet publiched :)) where and have their common meaning.

So QED

**Results**

- a very importand result.
- Something obvious,

**Exercises**

- If in a triangle ABC, A=45 and b=R=1, where R is the radious of the circumcircle of the triangle, find side c.
- If A, B, C, D are concyclic and line AB cuts CD at E, prove that . (You will need the power of a point theorem)