\(AC=a\sqrt{2}=SA\Rightarrow\Delta SAC\) vuông cân \(\Rightarrow SC=SA\sqrt{2}=2a\)
Kẻ \(AM\perp SC\Rightarrow M\) là trung điểm SC \(\Rightarrow AM=\dfrac{1}{2}SC=a\)
Gọi N là trung điểm AM \(\Rightarrow ON||CM\) (đường trung bình) \(\Rightarrow ON\perp AM\) \(\Rightarrow ON\perp\left(AHK\right)\)
\(\Rightarrow ON=d\left(O;\left(AHK\right)\right)\) ; \(ON=\dfrac{1}{2}CM=\dfrac{1}{4}SC=\dfrac{a}{2}\)
\(\dfrac{1}{AH^2}=\dfrac{1}{SA^2}+\dfrac{1}{AB^2}\Rightarrow AH=AK=\dfrac{a\sqrt{6}}{3}\)
Gọi I là giao điểm AM và HK
\(AK^2=AI.AM\Rightarrow AI=\dfrac{AK^2}{AM}=\dfrac{2a}{3}\)
\(IK=\sqrt{AK^2-AI^2}=\dfrac{a\sqrt{2}}{3}\Rightarrow HK=2IK=\dfrac{2a\sqrt{2}}{3}\)
\(V=\dfrac{1}{3}ON.\dfrac{1}{2}AI.HK=\dfrac{a^3\sqrt{2}}{27}\)
