Find the coordinates in an isosceles triangle if the triangle it self is in positive axis A at $(45,10)$,  B at $(10,20)$,  $AB=AC$ and angle $C=20$ degree find the coordinates of $C$.suggest the formula so i can write code in Perl.
 A: We have $A(45,10),B(10,20),C(x_c,y_c)$.
$AB=AC=\sqrt{(45-10)^{2}+(10-20)^{2}}=5\sqrt{53}$
$C=B=20^{%
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}=\pi /9$ rad
$A=180^{%
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}-40^{%
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}=140^{%
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}=7\pi /9$ rad

Let's make the following change of variables: $x=X+45,y=Y+10$ (translation of axes). Then $A$ becomes the origin of the $XY$ referential.
The vector $\overrightarrow{AB}$ can be written in this $XY$ referential as
$\overrightarrow{AB}=(5\sqrt{53}\cos \left( \pi -\arctan \frac{2}{7}\right)
,5\sqrt{53}\sin \left( \pi -\arctan \frac{2}{7}\right) )=(-35,10)$
and the vector $\overrightarrow{AC}$ as 
$\overrightarrow{AC}=(5\sqrt{53}\cos \left( \pi -\frac{7\pi }{9}-\arctan 
\frac{2}{7}\right) ,5\sqrt{53}\sin \left( \pi -\frac{7\pi }{9}-\arctan \frac{%
2}{7}\right) )$
Therefore in the original referencial $xy$, we have
$x_{C}=5\sqrt{53}\cos \left( -\arctan \frac{2}{7}-\frac{7\pi }{9}+\pi
\right) +45\approx 78.239$
$y_{C}=5\sqrt{53}\sin \left( -\arctan \frac{2}{7}-\frac{7\pi }{9}+\pi
\right) +10\approx 24.837$
A: You have the coordinates of A and B, so you can compute the distance AB.  AB=AC, so you then know the distance AC.  Let the coordinates of C be (x,y).  Apply the distance formula to A and C and set the result equal to the distance you already computed.  This equation guarantees that AB=AC.
Now, the angle at C is determined by the vectors $\overrightarrow{CA}$ and $\overrightarrow{CB}$.  These vectors can be found by subtracting the coordinates of C from A and B (respectively).  $(CA)(CB)\cos C = \overrightarrow{CA}\cdot\overrightarrow{CB}$, and you know the distances CA = AC = AB and the measure of C, and you can compute CB and the dot product in terms of (x,y).  This equation guarantees that the measure of C is 20°.
Solve the system resulting from the two equations above to find the coordinates of C.  There are almost certainly 2 solutions.
(See also this question.)
