Exercise 3 — Part A
In a plane P, consider a triangle ABC right-angled at A. Let $d$ be the line orthogonal to plane P and passing through point B. Consider a point D on this line distinct from point B.
1. Show that the line (AC) is orthogonal to the plane (BAD).
A bicoin is called a tetrahedron whose four faces are right triangles.
2. Show that the tetrahedron ABCD is a bicoin.
3. a. Justify that the edge $[CD]$ is the longest edge of the bicoin ABCD.
b. Let I be the midpoint of edge $[CD]$. Show that point I is equidistant from the 4 vertices of the bicoin ABCD.

As shown in the figure, $D$ is the apex of the cone, $O$ is the center of the base of the cone, $\triangle A B C$ is an equilateral triangle inscribed in the base, and $P$ is a point on $D O$ with $\angle A P C = 90 ^ { \circ }$ .
(1) Prove that plane $P A B \perp$ plane $P A C$ ;
(2) Given $D O = \sqrt { 2 }$ and the lateral surface area of the cone is $\sqrt { 3 } \pi$ , find the volume of the triangular pyramid $P - A B C$ .

As shown in the figure, in the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, points $E , F$ are on edges $D D _ { 1 } , B B _ { 1 }$ respectively, with $2 D E = E D _ { 1 } , B F = 2 F B _ { 1 }$. Prove:
(1) When $A B = B C$, $E F \perp A C$;
(2) Point $C _ { 1 }$ lies in plane $A E F$.

If the lines $x = ay + b,\, z = cy + d$ and $x = a'z + b',\, y = c'z + d'$ are perpendicular, then
(1) $cc' + a + a' = 0$
(2) $aa' + c + c' = 0$
(3) $bb' + cc' + 1 = 0$
(4) $ab' + bc' + 1 = 0$

In space, there is a regular cube $ABCD-EFGH$, where vertices $A, B, C, D$ lie on the same plane, and $\overline{AE}$ is one of its edges, as shown in the figure. Among the following options, select the plane that is perpendicular to both plane $BGH$ and plane $CFE$.
(1) Plane $ADH$
(2) Plane $BCD$
(3) Plane $CDG$
(4) Plane $DFG$
(5) Plane $DFH$