Question

Can someone help me with this task? ABCD is a parallelogram. Let E be the center of AB and let F be the center of the CD. Pull the lines AF and CE. Show that AF and CE divide the diagonal BD into three equal parts.

Answer 1

see explanation.

Explanation:

Given that `ABCD` is a parallelogram,
`=> AB=CD, AB "//" CD, AD=BC, AD "//" BCand angleABD=angleCDB=alpha`
as `E and F` are the midpoint of `AB and CD`, repectively,
`=> AF "//" EC`,
draw line `QM`, parallel to `DC`, as shown in the figure,
`=> anglePQM=alpha`,
let `angleDQF=beta, => angleQPM =angleBPE=beta`,
let `AE=y, => EB=DF=FC=QM=y`,
as `DF=QM=BE=y`, and the three corresponding angles of `DeltaDQF, DeltaQPM and DeltaBPE` are equal,
`=> DeltaDQF, DeltaQPM and DeltaBPE` are congruent,
Hence, `PB=PQ=QD`