求CD的长度？

hejoseph

\begin{align*}
&AB=\frac{\sin C}{\sin A}a，AC=\frac{\sin B}{\sin A}a\\
&AE_1=AB\sin B=\frac{\sin B\sin C}{\sin A}a，BE_1=AB\cos B=\frac{\cos B\sin C}{\sin A}a，CE_1=AC\cos C=\frac{\sin B\cos C}{\sin A}a\\
&DE_2=kAE_1=\frac{k\sin B\sin C}{\sin A}a\\
&BM=MC=\frac{a}{2}\\
&\angle BOM=\angle MOC=\angle D\\
&OB=OC=OD=\frac{a}{2\sin D}\\
&OM=OB\cos\angle BOM=\frac{a}{2}\cot D\\
&DP=DE_2-OM=\frac{k\sin B\sin C}{\sin A}a-\frac{a}{2}\cot D\\
&OP=\sqrt{OD^2-DP^2}=\sqrt{\left(\frac{a}{2\sin D}\right)^2-\left(\frac{k\sin B\sin C}{\sin A}a-\frac{a}{2}\cot D\right)^2}=\frac{\sqrt{1+4k\csc A\sin B\sin C(\cot D-k\csc A\sin B\sin C)}}{2}a\\
&BE_2=BM+OP=\frac{1+\sqrt{1+4k\csc A\sin B\sin C(\cot D-k\csc A\sin B\sin C)}}{2}a\\
&AE_3=BE_2-BE_1=\frac{-2\csc A\sin B\sin C+1+\sqrt{1+4k\csc A\sin B\sin C(\cot D-k\csc A\sin B\sin C)}}{2}a\\
&\triangle AEE_1\sim\triangle DAE_3\Rightarrow EE_1=\frac{DA}{AE}AE_3=\frac{1}{k-1}AE_3=\frac{-2\csc A\cos B\sin C+1+\sqrt{1+4k\csc A\sin B\sin C(\cot D-k\csc A\sin B\sin C)}}{2(k-1)}a\\
&\overline{BE}=BE_1-EE_1=\frac{2k\csc A\cos B\sin C-1-\sqrt{1+4k\csc A\sin B\sin C(\cot D-k\csc A\sin B\sin C)}}{2(k-1)}a\\
&\overline{EC}=CE_1+EE_1=\frac{2k\csc A\sin B\cos C-1+\sqrt{1+4k\csc A\sin B\sin C(\cot D-k\csc A\sin B\sin C)}}{2(k-1)}a
\end{align*}
另一组公式方法一样

Sirius

aimisiyou  可以写一下过程吗？看图没看出怎么做？