Question

Cho hình chóp \(SABCD\) gọi \(H,K\) lần lượt là trọng tâm \(\Delta SAB,\Delta SBC\) . M là trung điểm \(AC,I\in SM\) sao cho \(SI>SM\) .Tìm giao tuyến \(a,\left(IHK\right);\left(ABC\right)\\ b,\left(IHM\right);\left(SBC\right)\)

Answer 1

Gọi Q là trung điểm AB

Trong mp(IHS), gọi \(P=MQ\cap IH\)

a) Ta có:

\(\left\{{}\begin{matrix}P\in IH\subset\left(IHK\right)\\P\in MQ\subset\left(ABC\right)\end{matrix}\right.\)\(\Rightarrow P\in\left(IHK\right)\cap\left(ABC\right)\)

Lại có:

\(\left\{{}\begin{matrix}HK\text{/}\text{/}AC\left(Thales\right)\\HK\subset\left(IHK\right)\\AC\subset\left(ABC\right)\\\left(IHK\right)\cap\left(ABC\right)=d\end{matrix}\right.\)\(\Rightarrow d\text{/}\text{/}HK\text{/}\text{/}AC\)

\(\Rightarrow\left(IHK\right)\cap\left(ABC\right)=d\) đi qua P và \(d\text{/}\text{/}HK\text{/}\text{/}AC\)

b) Ta có:

\(\left\{{}\begin{matrix}S\in IM\subset\left(IHM\right)\\S\in\left(SBC\right)\end{matrix}\right.\)\(\Rightarrow S\in\left(IHM\right)\cap\left(SBC\right)\)

Lại có:

\(\left\{{}\begin{matrix}QM\text{/}\text{/}BC\left(Thales\right)\\QM\subset\left(IHM\right)\\BC\subset\left(SBC\right)\\\left(IHM\right)\cap\left(SBC\right)=d\text{'}\end{matrix}\right.\)\(\Rightarrow d\text{'}\text{/}\text{/}QM\text{/}\text{/}BC\)

\(\Rightarrow\left(IHM\right)\cap\left(SBC\right)=d\text{'}\) đi qua S và \(d\text{'}\text{/}\text{/}QM\text{/}\text{/}BC\)

Answer 2