a) Tứ giác AHIK nội tiếp

$\widehat{AHI}+\widehat{AKI}=90^0+90^0=180^0$

$\Rightarrow\text{AHIK nội tiếp}$

b) $IA\times IC=IB\times ID$

$\text{Xét }\Delta IAB\text{ và }\Delta IDC\text{ có:}$

$\widehat{AIB}=\widehat{DIC}\left(\text{2 góc đối đỉnh}\right)$

$\widehat{A_1}=\widehat{D_1}\left(\text{cùng chắn }\stackrel\frown{BC}\right)$

$\Rightarrow\Delta IAB\sim\Delta IDC\left(g-g\right)$

$\Rightarrow\dfrac{IA}{ID}=\dfrac{IB}{IC}$

$\Rightarrow IA\times IC=IB\times ID$

c) $\Delta HKI\sim\Delta BDC$

$\widehat{H_2}=\widehat{A_2}\left(\text{AHIK nội tiếp}\right)$

$\widehat{A_2}=\widehat{B_2}\left(\text{cùng chắn }\stackrel\frown{CD}\right)$

$\Rightarrow\widehat{H_2}=\widehat{B_2}$ (1)

$\widehat{K_1}=\widehat{A_1}\left(\text{AHIK nội tiếp}\right)$

$\widehat{A_1}=\widehat{D_1}\left(\text{cùng chắn }\stackrel\frown{BC}\right)$

$\Rightarrow\widehat{K_1}=\widehat{D_1}$ (2)

Từ (1) và (2) $\Rightarrow\Delta HKI\sim\Delta BDC\left(g-g\right)$

d) $\dfrac{S_{HKI}}{S_{ABD}}\le\dfrac{HK^2}{4AI^2}$

➤ $\Delta HKI\sim\Delta BDC\Rightarrow\dfrac{S_{HKI}}{S_{BDC}}=\dfrac{HK^2}{BD^2}\Rightarrow S_{HKI}=\dfrac{HK^2\times S_{BDC}}{BD^2}$

➤ $\text{Đặt }T=\dfrac{S_{HKI}}{S_{ABD}}=\dfrac{HK^2\times S_{BDC}}{BD^2\times S_{ABC}}$

Ta có: $\dfrac{S_{BDC}}{S_{ABC}}=\dfrac{IC}{IA}$

$\Rightarrow T=\dfrac{HK^2\times IC}{BD^2\times IA}=\dfrac{HK^2\times IC}{\left(IB+ID\right)^2\times IA}$

➤ Áp dụng bất đẳng thức AM - GM

$\Rightarrow T\le\dfrac{HK^2\times IC}{4\times IB\times ID\times IA}=\dfrac{HK^2\times IC}{4\times IA\times IC\times IA}=\dfrac{HK^2}{4IA^2}\left(đpcm\right)$