Solving triangle from two angles at the median and one side

[poolmich]

I have the following problem. I have a triangle and know the length of one side a. I also know the two angles alpha and beta that are formed by the median x of this side a. I now want to calculate b and c.

My first question was, whether this triangle is uniquely defined by these three values. And cannot prove it, but I think it should be. My second question then was how to calculate b, c and x. If I have one of them, I know how to get to the rest.



I started with all sine-rules and figured out, that

b*sin(alpha) = c*sin(beta)

I went through cosine rules but couldn't simplify the formulas anymore.

Is there a simple approach that I am missing?

thanks, Michael

 

[Dr.Peterson]

I have the following problem. I have a triangle and know the length of one side a. I also know the two angles alpha and beta that are formed by the median x of this side a. I now want to calculate b and c.

My first question was, whether this triangle is uniquely defined by these three values. And cannot prove it, but I think it should be. My second question then was how to calculate b, c and x. If I have one of them, I know how to get to the rest.

View attachment 10939

I started with all sine-rules and figured out, that

b*sin(alpha) = c*sin(beta)

I went through cosine rules but couldn't simplify the formulas anymore.

Is there a simple approach that I am missing?

thanks, Michael

I can prove geometrically that the triangle is uniquely determined by a, alpha, and beta; I did so by constructing it geometrically (by intersecting two circles).

If you just write the law of cosines applied to angle alpha+beta, you can easily solve for b or c as a function of a, alpha, and beta, though the result is not very pretty.

 

[poolmich]

Thanks

Thank you for the answer, Dr. Peterson!

It seems I was quite close already. I figured it out finally. The formula is not even that bad in the end.

 

[poolmich]

Thanks

Thanks for helping me!

It seems, I was already quite close. The formula is not too bad in the end (less so then I had scribbled, before I posted this)

In case anyone else also wants to know the answer:
Equation (1) I figured out before from the sine rules of the individual triangles, separated by the median,
equation (2) is the cosine rule for a and (alpha+beta) and in
equation (3) x is the length of the median of a.
First solve b then c then x.