<s>[INST] Coordinate of the excentre of a triangle I am just wondering that how the coordinate of the excentre comes out if we know the coordinates of vertices of the triangle.
 [/INST] Consider the diagram:

$d=\overline{CE}=\overline{CF}$. Note that $c=\overline{AB}=(d-a)+(d-b)$. Therefore,
$$
d=\frac{a+b+c}2\tag{1}
$$
Furthermore, $d=\overline{CD}\cos(\theta/2)$ and $\overline{CH}=d\cos(\theta/2)$; therefore, $\overline{CH}=\overline{CD}\cos^2(\theta/2)$.
The Law of Cosines gives
$$
\cos(\theta)=\frac{a^2+b^2-c^2}{2ab}\tag{2}
$$
so that
$$
\cos^2(\theta/2)=\frac{\vphantom{b^2}1+\cos(\theta)}2=\frac{(a+b)^2-c^2}{4ab}\tag{3}
$$
$H$ is the mid-point of $\overline{EF}$; therefore,
$$
\begin{align}
H-C
&=\frac{d}2\left(\frac{B-C}a+\frac{A-C}b\right)\\[6pt]
&=\cos^2(\theta/2)(D-C)\tag{4}
\end{align}
$$
Putting together $(1)$, $(3)$, and $(4)$, we get
$$
\begin{align}
D
&=C+\frac{4ab}{(a+b)^2-c^2}\frac{a+b+c}4\left(\frac{B-C}a+\frac{A-C}b\right)\\
&=C+\frac{ab}{a+b-c}\left(\frac{B-C}a+\frac{A-C}b\right)\\
&=C+\frac{a(A-C)+b(B-C)}{a+b-c}\\[6pt]
&=\frac{aA+bB-cC}{a+b-c}\tag{5}
\end{align}
$$
</s>