I have a picture:

From point M, draw ray MN parallel with AC. (N\(\in\)DB)

AB=AD \(\Rightarrow\)\(\Delta\)BAD is an isosceles triangle \(\Rightarrow\widehat{ABD}=\widehat{ADB}\)

But \(\widehat{MNB}=\widehat{ADB}\) (Isotopes)

So \(\widehat{ABD}=\widehat{MNB}\) or \(\widehat{MBN}=\widehat{MNB}\) 

\(\Rightarrow\Delta BMN\) is a isosceles triangle \(\Rightarrow BM=MN\).

Because BM=CD \(\Rightarrow MN=CD\).

\(MN\)//AC \(\Rightarrow\)MN//CD \(\Rightarrow\left\{{}\begin{matrix}\widehat{MNI}=\widehat{CDI}\\\widehat{NMI}=\widehat{DCI}\end{matrix}\right.\)

\(\Rightarrow\Delta MIN=\Delta CID\) (Angular angle)

Now we can prove that IM=IC (Corresponding edges)