Question 60
Three distinct points $X, Y$ and $Z$ are given in space; we ask whether there exists a point $P$ different from $X, Y$ and $Z$ such that the lines $P X, P Y$ and $P Z$ are pairwise perpendicular. Four friends make the following statements:
Alberto: "There exist $X, Y, Z$ and $P$ lying in the same plane that satisfy these conditions."
Barbara: "There do not exist 4 points $X, Y, Z$ and $P$ that satisfy these conditions."
Carlo: "For $P$ to exist it is necessary that $X Y Z$ be acute-angled."
Daria: "For $P$ to exist it is sufficient that $X Y Z$ not be obtuse-angled."
Who is right?
(A) Only Alberto
(B) Only Barbara
(C) Only Carlo
(D) Only Daria
(E) None of the above.
Alberto: "There exist $X, Y, Z$ and $P$ lying in the same plane that satisfy these conditions."
Barbara: "There do not exist 4 points $X, Y, Z$ and $P$ that satisfy these conditions."
Carlo: "For $P$ to exist it is necessary that $X Y Z$ be acute-angled."
Daria: "For $P$ to exist it is sufficient that $X Y Z$ not be obtuse-angled."
Who is right?
(A) Only Alberto
(B) Only Barbara
(C) Only Carlo
(D) Only Daria
(E) None of the above.