Track / Stage 3 / 161 of 260 #641 of 2444

Problem 641

AMC 10/12, early questions

Geometry Difficulty 3.7 Prove it Junior Macedonian Mathematical Olympiad · North Macedonia

Let $ABCD$ be a parallelogram and let $E$ , $F$ , $G$ and $H$ be the midpoints of the sides $AB$ , $BC$ , $CD$ and $DA$ , respectively. If $BH \cap AC = I$ , $BD \cap EC = J$ , $AC \cap DF = K$ and $AG \cap BD = L$ , then prove that the quadrilateral $IJKL$ is a parallelogram.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Let $AC \cap BD = O$ . Clearly, $AO$ and $BH$ are medians in the triangle $ABD$ , hence $I$ is the centroid of $ABD$ . Similarly $K$ is the centroid of $BCD$ . If $\overline{IO} = x$ , then $\overline{AI} = 2x$ . Similarly, if $\overline{KO} = y$ , then $\overline{CK} = 2y$ . Therefore $3x = \overline{AO} = \overline{CO} = 3y$ , i.e. $x = y$ . We analogously prove that $\overline{JO} = \overline{LO}$ . It follows that $IJKL$ is a parallelogram.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.