1988 CMO 第 6 题

Given three tetrahedrons $A_iB_i C_i D_i$ ( $i=1,2,3$ ), planes $\alpha _i,\beta _i,\gamma _i$ ( $i=1,2,3$ ) are drawn through $B_i ,C_i ,D_i$ respectively, and they are perpendicular to edges $A_i B_i, A_i C_i, A_i D_i$ ( $i=1,2,3$ ) respectively. Suppose that all nine planes $\alpha _i,\beta _i,\gamma _i$ ( $i=1,2,3$ ) meet at a point $E$ , and points $A_1,A_2,A_3$ lie on line $l$ . Determine the intersection (shape and position) of the circumscribed spheres of the three tetrahedrons.

给定三个四面体 $A_iB_i C_i D_i$ ( $i=1,2,3$ )，分别通过 $B_i ,C_i ,D_i$ 绘制平面 $\alpha _i,\beta _i,\gamma _i$ ( $i=1,2,3$ )，并且它们垂直于边 $A_i B_i, A_i C_i, A_i D_i$ ( $i=1,2,3$ ) 分别。假设所有九个平面 $\alpha _i,\beta _i,\gamma _i$ ( $i=1,2,3$ ) 相交于点 $E$ ，并且点 $A_1,A_2,A_3$ 位于直线 $l$ 上。确定三个四面体外接球的交点(形状和位置)。