Xét $\triangle MNP$ vuông tại $M$, $I$ là tâm nội tiếp

$\Rightarrow MI$ là phân giác $\widehat{M}$

$\Rightarrow \widehat{KMI} = \widehat{QMI} = 45^\circ$

Xét $\triangle MIK$:

$\widehat{MIK} = \widehat{IMN} + \widehat{INM} = 45^\circ + \dfrac{\widehat{N}}{2}$ (góc ngoài $\triangle MNI$)

Xét $\triangle MQI$:

$\widehat{MQI} = 180^\circ - \widehat{M} - \widehat{MPQ} = 90^\circ - \dfrac{\widehat{P}}{2}$

Mà $\dfrac{\widehat{P}}{2} = 45^\circ - \dfrac{\widehat{N}}{2} \Rightarrow \widehat{MQI} = 90^\circ - (45^\circ - \dfrac{\widehat{N}}{2}) = 45^\circ + \dfrac{\widehat{N}}{2}$

$\Rightarrow \widehat{MIK} = \widehat{MQI}$

Xét $\triangle MIK$ và $\triangle MQI$:

$\widehat{KMI} = \widehat{IMQ} = 45^\circ$

$\widehat{MIK} = \widehat{MQI}$ (cmt)

$\Rightarrow \triangle MIK \sim \triangle MQI$ (g.g)

$\Rightarrow \dfrac{MI}{MQ} = \dfrac{MK}{MI} \Rightarrow MI^2 = MK \cdot MQ$ (đpcm)