11.233 a) The lengths of the edges $AB, AC, AD$ and $BC$ of an orthocentric tetrahedron are 5, 7, 8, and 6 cm, respectively. Find the lengths of the remaining two edges. b) Is the tetrahedron $ABCD$ orthocentric if $AB=8$ cm, $BC=12$ cm, and $DC=6$ cm?
Solution. a) According to item v) of problem 11.232, $A B^{2}+C D^{2}=A C^{2}+B D^{2}=A D^{2}+B C^{2}=100 \Rightarrow C D=\sqrt{75}, B D=\sqrt{51}$.

Answer: $5 \sqrt{3}$ cm, $\sqrt{51}$.

b) If the tetrahedron were orthocentric, then

$A B^{2}+C D^{2}=100=B C^{2}+A D^{2} \Rightarrow A D^{2}=100-144<0$, which is impossible.