Solid Geometry – Problem 5: Prove that plane

In the quadrilateral pyramid $B-ACC_1A_1$, $AA_1\perp$ plane $ABC$, $CC_1\perp$ plane $ABC$, $BA\perp AC$, and $$AB=AC=CC_1=2AA_1.$$

(1) Prove that plane $A_1BC_1\perp BCC_1$.

(2) Let $P$ be the reflection of point $C$ across plane $A_1BC_1$. Find $\sin\theta$, where $\theta$ is the angle between line $AP$ and plane $A_1BC_1$.

(1) Let $M,N$ be the midpoints of $BC_1$, $BC$, respectively, and connect $A_1M,AN$. From $AA_1\parallel CC_1$, $AA_1=\dfrac12CC_1$, and $MN\parallel CC_1$, $MN=\dfrac12CC_1$, we get $AA_1\parallel MN$ and $AA_1=MN$. Hence quadrilateral $A_1MNA$ is a parallelogram, so $A_1M\parallel AN$.

Also, $AB=AC\Rightarrow AN\perp BC$, and $AN\perp CC_1$. Since $BC\cap CC_1=C$, we have $AN\perp$ plane $BCC_1$, thus $A_1M\perp$ plane $BCC_1$. Because $A_1M\subset$ plane $A_1BC_1$, $$\text{plane }A_1BC_1\perp\text{plane }BCC_1.$$

(2) Through $C$, draw $CQ\perp BC_1$, meeting $BC_1$ at $Q$. Then $CQ\perp$ plane $A_1BC_1$. Let $AB=AC=CC_1=2AA_1=2$. Set a 3D rectangular coordinate system with $A$ as origin, and lines containing $AB,AC,AA_1$ as the $x,y,z$-axes. Take $$B(2,0,0),\ C(0,2,0),\ C_1(0,2,2),\ A_1(0,0,1).$$ From right-triangle projection relations, $2C_1Q=QB$, so $$Q\left(\frac23,\frac43,\frac43\right).$$ Since $P$ is the reflection of $C$ across plane $A_1BC_1$, point $Q$ is the midpoint of $CP$, thus $$P\left(\frac43,\frac23,\frac83\right),\quad \overrightarrow{AP}=\left(\frac43,\frac23,\frac83\right).$$ A normal vector of plane $A_1BC_1$ can be $\vec n=(1,-1,2)$. Therefore $$\sin\theta=\frac{|\overrightarrow{AP}\cdot\vec n|}{|\overrightarrow{AP}|\,|\vec n|}=\frac{3\sqrt{14}}{14}.$$

(1) By midpoint construction and perpendicular-plane criteria, we find a line in plane $A_1BC_1$ that is perpendicular to plane $BCC_1$, so $$\text{plane }A_1BC_1\perp\text{plane }BCC_1.$$

(2) After locating the reflection point $P$ through midpoint $Q$, and using a normal vector of plane $A_1BC_1$, we obtain $$\sin\theta=\dfrac{3\sqrt{14}}{14}.$$ This is the sine of the angle between line $AP$ and the plane.