Question

Here's the problem
I have a scalene Triangle with Vertices A,B and C. I know the coordinates of A and B. and i know angle CAB and angle CBA. now I have to find the coordinates of C.
please help me with the Solution

Answer 1

Conditions

Here's the problem

I have a scalene Triangle with Vertices A,B and C. I know the coordinates of A and B. and I know angle CAB and angle CBA. now I have to find the coordinates of C.

please help me with the Solution

Solution

We must use vector interpretation of this problem.

Consider points A and B with known coordinates:

$$A(a_1, a_2), B(b_1, b_2)$$

And point C with unknown coordinates:

$$C(c_1, c_2),$$

Consider vectors:

$$\overline{AC} = (c_1 - a_1, c_2 - a_2)$$ $$\overline{BC} = (c_1 - b_1, c_2 - b_2)$$ $$\overline{AB} = (b_1 - a_1, b_2 - a_2)$$ $$\overline{BA} = (a_1 - b_1, a_2 - b_2)$$

We know, that the scalar product of two vectors is related to a cosine of an angel between these vectors in the following way:

$$\overline{AC} \cdot \overline{AB} = |\overline{AC}| \cdot |\overline{AB}| \cdot \cos CAB$$

And

$$\overline{AB} \cdot \overline{BC} = |\overline{AB}| \cdot |\overline{BC}| \cdot \cos CBA$$

You know the values of both angles; hence, you know their cosines. There are two unknown variables in these two equations - $c_1, c_2$. Put these equations to a system, solve it and you will get the coordinates of point C.