c) Ta có \(\Delta AHB\sim\Delta ADC(g.g)\) nên \(\dfrac{BH}{BA}=\dfrac{CD}{CA}\).
Lại có \(\dfrac{BI}{BH}=\dfrac{CN}{CD}\) nên \(\dfrac{BI}{BA}=\dfrac{CN}{CA}\).
Mà \(\widehat{IBA}=\widehat{NCA}\) nên \(\Delta IBA\sim\Delta NCA(c.g.c)\)
\(\Rightarrow \widehat{ANC}=\widehat{AIB}\)
\(\Rightarrow\widehat{AID}=\widehat{AND}\).
Mà \(\widehat{API}=\widehat{DPN}\) (đối đỉnh) nên \(\Delta API\sim\Delta DPN(g.g)\)
\(\Rightarrow\dfrac{PA}{PI}=\dfrac{PD}{PN}\).
Mà \(\widehat{APD}=\widehat{IPN}\) (đối đỉnh) nên \(\Delta APD\sim\Delta IPN(c.g.c)\)
\(\Rightarrow\widehat{PIN}=\widehat{PAD}\).
Ta có \(\widehat{AIN}=\widehat{AID}+\widehat{NID}=\widehat{NAD}+\widehat{AND}=90^o\) nên \(AI\perp NI\).
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