Parallelo-split

Age
14 to 16
Challenge level
filled star empty star empty star
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



$ABCD$ is a parallelogram.

A diagonal, $AC$, has been drawn, as well as two lines $ED$ and $FB$, from the midpoints of sides $AB$ and $CD$, respectively, to opposite vertices.

Parallelo-split


Draw a parallelogram and construct the lines shown above. You may wish to use GeoGebra.

What do you notice about the lengths $AG$, $GH$ and $HC$?

Is it true for all parallelograms?

Can you prove it?

Extension

Can you split the diagonals of other quadrilaterals in a similar way?