Question 513348: Proof that the diagonals of a parallelogram bisect eachother
Answer by solver91311(24713)   (Show Source):
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Theorem If ABCD is a parallelogram, then prove that the diagonals of ABCD bisect each other.
Proof
Let the two diagonals be AC and BD and O be the intersection point.
We have to prove that O is the midpoint of AC and also the midpoint of BD.
Hence,  and 
We will prove using congruent triangles concept.
Consider two Triangles ABO and COD.
1.  ....( Line AC is a transversal of the parallel lines AB and CD, hence alternate angles).
2.  ....(Line BD is a transversal of the parallel lines AB and CD, hence alternate angles).
3.  ....(Opposite angles when two lines intersect each other area equal)
From conditions 1,2 and 3
Triangle ABO is similar to triangle CDO (By Angle -Angle similar property)
Since Triangles are similar, Hence ratio of sides are equal from similar triangles property.
 .........(4)
From theorem that Opposite sides of a parallelogram are equal,
 ..........(5)
From equation (4) and (5)
Similarly, 
Hence, We conclude that AO = CO and BO = DO.
QED
John
My calculator said it, I believe it, that settles it
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