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”.

enter image description here

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.

Answer

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

A=p(pa)(pb)(pc)=ah2,

where h is the altitude length and 2p=a+b+c is the perimeter. A geometric proof of (1) can be found in this post, in Portuguese. Solving (1) for h we find that

h=2p(pa)(pb)(pc)a=2aa+b+c2(a+b+c2a)(a+b+c2b)(a+b+c2c)=12a(a+b+c)(b+ca)(a+cb)(a+bc)

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 (xB,yB)=(0,0), then the coordinates of the upper vertex are

(xA,yA)=(c2h2,h).

The coordinates of the foot of the altitude h are thus (xN,yN)=(c2h2,0).

enter image description here

Attribution
Source : Link , Question Author : Dr. Mathematician , Answer Author : Américo Tavares

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