$\def\A{{\bf A}}
\def\B{{\bf B}}
\def\C{{\bf C}}
\def\R{{\bf R}}
\def\D{{\bf D}}
\def\f{\phi}$As others have mentioned, if the task is to find the coordinates of $B$ and $C$, this problem is underdetermined.
Let's first consider the triangle with point $A$ located at the origin and point $C$ lying along the positive $x$-axis.
Then the coordinates of the points can be found with simple trigonometry,
$$\begin{eqnarray*}
\A_0 &=& (0,0) \\
\B_0 &=& (c \cos A, c\sin A) \\
\C_0 &=& (b,0).
\end{eqnarray*}$$
The angle $A$, and the sides $b$ and $c$ have been given.
This is an SAS triangle.
The triangle can be solved by finding $a$ with the law of cosines and one of the other angles with the law of sines.
(Added: For completeness, $A = 72^\circ$, $b = 2.61r$, and $c=r$.
The law of cosines gives $a = \sqrt{b^2+c^2-2bc\cos A} = 2.49r$.
Then $\frac{\sin A}{a} = \frac{\sin B}{b}$ implies $B = 86^\circ$.
Lastly, $A+B+C = 180^\circ$ implies $C = 22^\circ$.)
The collection of triangles you are interested in have vertices of the form
$$\begin{equation*}
\D = \A + \R(\f)\D_0 \tag{1}
\end{equation*}$$
where $\A = (A_x,A_y)$ is the given location of $A$, and where
$\R(\f)$ is a rotation matrix.
The transformation (1) is a counterclockwise rotation by the angle $\f$, followed by a shift so the point $A$ has the given coordinates.
In components,
$$\begin{eqnarray*}
A_x &=& A_x \\
A_y &=& A_y \\
B_x &=& A_x + c\cos A\cos\f - c\sin A\sin\f \\
&=& A_x + c\cos(A+\f) \\
B_y &=& A_y + c\cos A\sin\f + c\sin A\cos\f \\
&=& A_y + c\sin(A+\f) \\
C_x &=& A_x + b\cos \f \\
C_y &=& A_y + b\sin \f.
\end{eqnarray*}$$
Below we plot the triangle before rotation and translation in black.
(We set $r = 1$ in the figure.)
The dotted triangle has been rotated counterclockwise by $\f = 30^\circ$, and then translated so the new location of point $A$ is $(2,1)$.
For reference,
$$\begin{eqnarray*}
A_x &=& 2 \\
A_y &=& 1 \\
c &=& r = 1 \\
b &=& 2.61 \\
A &=& 72^\circ \\
\f &=& 30^\circ.
\end{eqnarray*}$$
Let's look at a simpler problem. Suppose you have the situation depicted in the figure below:
Then, given the angle $\alpha$, the coordinates of the point $C''$ are:
$$
C''_x = r\cos\alpha
\qquad\mbox{and}\qquad
C''_y = r\sin\alpha
$$
where $r$ is the radius of the circle.
Now let's look at a slightly more complicated problem, depicted below:
This is very similar to the situation above. In fact,
$$
C'_x = r\cos(\alpha+\beta)
\qquad\mbox{and}\qquad
C'_y = r\sin(\alpha+\beta)
$$
By using the trigonometric relations $\sin(\alpha+\beta) = \sin\alpha\cos\beta + \sin\beta\cos\alpha$ and $\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$, we can write the above as follows:
$$
C'_x = r\cos\alpha\cos\beta - r\sin\alpha\sin\beta
\qquad\mbox{and}\qquad
C'_y = r\sin\alpha\cos\beta + r\sin\beta\cos\alpha
$$
But, wait... By looking at the previous situation and replacing $C''$ with $B'$ and $\alpha$ with $\beta$, we see that
$$
B'_x = r\cos\beta
\qquad\mbox{and}\qquad
B'_y = r\sin\beta
$$
Therefore, we can write
$$
C'_x = B'_x\cos\alpha - B'_y\sin\alpha
\qquad\mbox{and}\qquad
C'_y = B'_x\sin\alpha + B'_y\cos\alpha
$$
But what you want is this, instead:
Well, we can just move everything rigidly by the vector $-\vec{OA}$ so that $A$ is now the origin of the coordinate system and we get the situation just above. This amounts to subtracting $A$ from both $B$ and $C$ to get $B'$ and $C'$ in the above, and we find
$$
C_x - A_x = (B_x-A_x)\cos\alpha - (B_y-A_y)\sin\alpha
$$
$$
C_y - A_y = (B_x-A_x)\sin\alpha + (B_y-A_y)\cos\alpha
$$
Then, finally,
$$
C_x = A_x + (B_x-A_x)\cos\alpha - (B_y-A_y)\sin\alpha
$$
$$
C_y = A_y + (B_x-A_x)\sin\alpha + (B_y-A_y)\cos\alpha
$$
Best Answer
Consider the angle formed between line ($AC$) and the x-axis. The picture you provided shows point $B$ on the x-axis which is not necessarily the case. The x-distance from point $A$ would be $\text{radius}\times \sin(\text{angle})$. The y-distance from point $A$ would be $\text{radius}\times \cos(\text{angle})$