Euclidean Geometry

Prove the Ceva's theorem first :
If the lines AD, BE, and CF pass through a common point , then
AF/FB * BD/DC * CE/EA = 1
Proof :AF/FB * BD/DC * CE/EA
= △AOC / △BOC * △AOB / △AOC * △BOC / △AOB
= 1***************************************************************************
Then prove the converse theorem :
圖片參考：http://imgcld.yimg.com/8/n/HA04628698/o/701201180040913873421520.jpg

If AF/FB * BD/DC * CE/EA = 1 , then
the lines AD, BE, and CF pass through a common point .
Proof :Let AD and BE meet at O , then join CO meet AB at F' :By the Ceva's theorem we have :
AF'/F'B * BD/DC * CE/EA = 1On the other hand ,
AF/FB * BD/DC * CE/EA = 1 (given)Hence,AF'/F'B = AF/FB
AF'/F'B + 1 = AF/FB + 1
(AF' + F'B) / F'B = (AF + FB) / FB
AB / F'B = AB / FB
F'B = FBThat means F' is F , therefore AD, BE, and CF pass through a common point .
Q.E.D.