Question 56
A point $P$ inside a triangle $ABC$ is called exceptional if the following condition holds:
If $A'$, $B'$ and $C'$ are the intersections of the lines $AP$, $BP$ and $CP$ with the sides $BC$, $AC$ and $AB$, respectively, then the segments $AA'$, $BB'$ and $CC'$ can be translated (moved parallel to themselves) so that they form a triangle whose side lengths are $|AA'|$, $|BB'|$ and $|CC'|$.
a. Prove that the centroid of the triangle is an exceptional point.
b. Prove the statement "The only exceptional point inside a triangle is the centroid of the triangle."
If $A'$, $B'$ and $C'$ are the intersections of the lines $AP$, $BP$ and $CP$ with the sides $BC$, $AC$ and $AB$, respectively, then the segments $AA'$, $BB'$ and $CC'$ can be translated (moved parallel to themselves) so that they form a triangle whose side lengths are $|AA'|$, $|BB'|$ and $|CC'|$.
a. Prove that the centroid of the triangle is an exceptional point.
b. Prove the statement "The only exceptional point inside a triangle is the centroid of the triangle."