a, \(\widehat{CPK}=\widehat{CPI}=90^o=\widehat{KBC}\) (Góc nội tiếp chắn nửa đường tròn)

\(\Rightarrow CPKB\) nội tiếp đường tròn đường kính \(CK\)

\(\Rightarrow\) Tâm là trung điểm CK

b, \(\left\{{}\begin{matrix}\widehat{KBC}=\widehat{CAI}=90^o\\\widehat{KCB}=\widehat{CIA}\left(\text{phụ góc ICA}\right)\end{matrix}\right.\)

\(\Rightarrow\Delta KBC\sim\Delta CAI\left(g-g\right)\)

\(\Rightarrow AI.BK=AC.BC\)

c, \(\widehat{KPB}=\widehat{KCB}=\widehat{CIA}=\widehat{CPA}\)

\(\Rightarrow\widehat{APB}=\widehat{CPA}+\widehat{CPB}=\widehat{KPB}+\widehat{CPB}=90^o\)

d, \(AI.BK=AC.BC\Rightarrow BK=\frac{AC.BC}{AI}\le\frac{\left(AC+BC\right)^2}{4AI}=\frac{AB^2}{4AI}\)

\(S_{ABKI}=\frac{1}{2}\left(AI+BK\right)AB\le\frac{1}{2}\left(AI+\frac{AB^2}{4AI}\right).AB\)

\(Max_{S_{ABKI}}=\frac{1}{2}\left(AI+\frac{AB^2}{4AI}\right).AB\Leftrightarrow AC=BC\)

\(\Rightarrow\) C là trung điểm AB