Question 58
A convex quadrilateral $ABCD$ is given. Let $E$ be the intersection of $AB$ and $CD$, $F$ be the intersection of $AD$ and $BC$, and $G$ be the intersection of $AC$ and $EF$. Prove that the following two statements are equivalent:
(i) $BD$ and $EF$ are parallel
(ii) $G$ is the midpoint of the segment $\overline{EF}$
Macedonian source version, translated in full:
A convex quadrilateral $ABCD$ is given. Let $E$ be the intersection of $AB$ and $CD$, $F$ be the intersection of $AD$ and $BC$, and $G$ be the intersection of $AC$ and $EF$. Prove that the following two statements are equivalent:
(i) $BD$ and $EF$ are parallel
(ii) $G$ is the midpoint of the segment $\overline{EF}$
(i) $BD$ and $EF$ are parallel
(ii) $G$ is the midpoint of the segment $\overline{EF}$
Macedonian source version, translated in full:
A convex quadrilateral $ABCD$ is given. Let $E$ be the intersection of $AB$ and $CD$, $F$ be the intersection of $AD$ and $BC$, and $G$ be the intersection of $AC$ and $EF$. Prove that the following two statements are equivalent:
(i) $BD$ and $EF$ are parallel
(ii) $G$ is the midpoint of the segment $\overline{EF}$