MHB How to find angles of a triangle

[Andrei1]

We know that in triangle ABC angle A equals $$\alpha$$ and side $$a=\frac{b+c}{2}.$$ How to find angles B and C knowing that $$B\geqslant C$$? For which values of $$\alpha$$ the problem has solutions?

ps. a, b, c are only notations.

answer. $$\frac{\pi-\alpha}{2}\pm\arccos(2\sin\frac{\alpha}{2})$$

 

[MarkFL]

Re: how to find angles of a triangle

It seems you have given $\alpha$ in terms of $\alpha$.

 

[Andrei1]

Re: how to find angles of a triangle

Here is what I do. I express a through b, c and $$\alpha$$ by cosine theorem. So I have an equation with only b and c as unknown. I obtain $$\frac{b}{c}$$ by sine theorem and I substitute this ratio in the first equation. I also know that $$\alpha=\pi-B-C.$$ So I have two equations with two unknowns. But it is hard to solve.

I also noticed that the bisector of angle A divide side a in two segments: b/2 and c/2.

 

[Opalg]

Re: how to find angles of a triangle

Here is what I do. I express a through b, c and $$\alpha$$ by cosine theorem. So I have an equation with only b and c as unknown. I obtain $$\frac{b}{c}$$ by sine theorem and I substitute this ratio in the first equation. I also know that $$\alpha=\pi-B-C.$$ So I have two equations with two unknowns. But it is hard to solve.

I also noticed that the bisector of angle A divide side a in two segments: b/2 and c/2.

I think you are nearly there. Look at this picture, in which $AD$ is the angle bisector at $A$, and $BN$ is perpendicular to $AD$:

​

The angle at $B$ is obviously $\angle ABN + \angle NBD$. Equally obviously, $\angle ABN = \frac{\pi}2 - \frac\alpha2$, so we just need to show that $\cos(\angle NBD) = 2\sin\bigl(\frac\alpha2\bigr)$. But $\cos(\angle NBD) = \frac{BN}{BD}$, and you have already shown that $BD = c/2$, so you just need to observe that $BN = c\sin\bigl(\frac\alpha2\bigr)$, which is evident from the triangle $ABN$.

You can get the result for the angle at $C$ in a similar way by dropping a perpendicular from $C$ to the extension of $AD$.​