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2006 Romanian NMO Problems/Grade 8/Problem 1

Problem

We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals.

Solution

We use the lemma that given two non-coplanar circles in space that intersect at two points, there exists a point P such that P is equidistant from any point on one circle and any point on the other circle.

Proof of lemma: Let the two circles be $O_1$ and $O_2$ and let them intersect at $X$ and $Y$. Draw the line $l_1$ through (the center of) $O_1$ perpendicular to the plane of the circle. We know that any point on that line is equidistant from any point on $O_1$ because for a point $P$ on the line and a point $Q$ on the circle, $PQ=\sqrt{PO_1^2+O_1Q^2}=\sqrt{PO_1^2+r_1^2}$, which does not depend on the point $Q$ chosen. Similarly, we draw the line $l_2$ through $O_2$. Since $X$ and $Y$ lie on both circles, any point on $l_1$ or $l_2$ is equidistant from $X$ and $Y$. However, the locus of all points equidistant from $X$ and $Y$ is the plane that perpendicularly bisects $\overline{XY}$. Therefore, $l_1$ and $l_2$ lie on one plane. Since our two circles are not coplanar, $l_1$ and $l_2$ must intersect at our desired point.

Let the cube have vertices $A$, $B$, $C$, $D$, $A'$, $B'$, $C'$, $D'$, where all sides but $A'B'C'D'$ are known to be cyclic quadrilaterals. First, we consider the circumcircles of quadrilaterals $ABCD$ and $ABB'A'$. By our lemma, there exists a point $O$ equidistant from $A$, $B$, $A'$, $B'$, $C$, $D$. Let the perpendicular from $O$ to the plane $AA'D'D$ intersect the plane at $O'$. By HL congruency, the triangles $OO'A$, $OO'A'$, and $OO'D$ are congruent. Since $O'A=O'A'=O'D$, O is the center of quadrilateral $AA'D'D$. By SAS congruency, $\triangle OO'D'$ is congruent to the aforementioned triangles, so $OA'=OD'$. Similarly, if we focus on quadrilateral $BCC'B'$, we get that $OB'=OC'$. Therefore, $OA'=OB'=OC'=OD'$. Let the perpendicular from $O$ to the plane $A'B'C'D'$ intersect the plane at $O''$. By HL congruency, the triangles $OO''A'$, $OO''B'$, $OO''C'$, and $OO''D'$ are congruent. Thus, $O''A'=O''B'=O''C'=O''D'$ and $A'B'C'D'$ is cyclic.

See also

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