Question 3
Let $ABC$ be a triangle and let $d$ pass through $A$ without being parallel to $BC$.
(a) Construct parallelogram $CEFG$ with $E,F,G$ respectively on the lines $d,AB,BC$, and $CE$ parallel to the median $AI$. Then construct parallelogram $EAKH$ with $K,H$ respectively on $AB,BC$. Put $U=BC\cap d$ and let $V$ be the reflection of $U$ in $A$. Prove that $V,I,FG\cap KH$ are collinear, and also collinear with $B',C'$, the other two vertices of the parallelogram $BB'CC'$ whose opposite vertices are $B,C$ and whose successive sides are parallel to $d,AI$.
(b) How are the pairs of intersection points of $FG,KH$ with the three lines $BC,CA,AB$ arranged? State a general theorem. Given a point $O$ in the plane of $ABC$, can two such lines $FG,KH$ be constructed through $O$?
(a) Construct parallelogram $CEFG$ with $E,F,G$ respectively on the lines $d,AB,BC$, and $CE$ parallel to the median $AI$. Then construct parallelogram $EAKH$ with $K,H$ respectively on $AB,BC$. Put $U=BC\cap d$ and let $V$ be the reflection of $U$ in $A$. Prove that $V,I,FG\cap KH$ are collinear, and also collinear with $B',C'$, the other two vertices of the parallelogram $BB'CC'$ whose opposite vertices are $B,C$ and whose successive sides are parallel to $d,AI$.
(b) How are the pairs of intersection points of $FG,KH$ with the three lines $BC,CA,AB$ arranged? State a general theorem. Given a point $O$ in the plane of $ABC$, can two such lines $FG,KH$ be constructed through $O$?