# Trigonometry, knowing 3 sides how to find the height?

I have a mathematician problem where, I knew the 3 sides of a triangle, with these sides I can figer out what type of type of triangle is.
What I realy want to find is the height of the triangle and another one “side”.

Let me explain what I want, with the above picture.
I knew a,b and c and I can calculate the 2 angles (the angle oposite of C, and the angle oposite of b, those on the black dots). I want to find out the red dots (the bottom red dot is is the other “side” I note before, from black dot to red dot distance).

So I want to to know the (x,y) for the red dot at the top, via the combination of height length, b OR c length (which also needs the two angles) in case to find the absolute right (x,y) for the red dot at the top.
Thanks.

Use Heron’s Formula for the area $A$ of a triangle

where $h$ is the altitude length and is the perimeter. A geometric proof of $(1)$ can be found in this post, in Portuguese. Solving $(1)$ for $h$ we find that

how this will help me to find the (x,y) for the top red dot in the
picture?

Concerning coordinates, if the base $a$ is horizontal and the coordinates of the left vertex are $(x_{B},y_{B})=(0,0)$, then the coordinates of the upper vertex are

The coordinates of the foot of the altitude $h$ are thus