___________________________________|______________
1.  D midpoint side AB             |  G
2.  F midpoint side AC             |  G
3.  segment ED median              |  def of median
4.  segment BF median              |  def of median
5.  segment AE median              |  only one line concurrunt with other median
6.  N point of concurrency of medians triangle ABC  |  def of point of concurrency
7.  segment NE / segment AN = 1/2  |  median thm
8.  point O point of concurrency triangle ABE  |  def of point of concurrency
9.  (segment BP / segment PE) * (segment EN / segment NA) * (segment AD / segment DB) = 1  |  cheva
10. segment AD congruent segment DB  |  def of midpoint
11. m segment AD = m segment DB    |  docs 10
12. segment AD / segment DB        |  ?
13. segment AD / segment AD = 1    |  sub 11 , 12
14. (segment BP / segment PE) * (1/2) * (1) = 1  |  sub 7, 13, 9
15. segment BP / (segment PE * 2) = 1  |  multiplication 14
16. segment BP = segment PE * 2    |  cross multipy 15
17. segment BP / segment PE = 2/1  |  division 16