Prove that this figure is a rhombus

Apr 2010
118
6
I am working through Serge Lang's book on geometry, and I am having trouble for some reason on this proof:

Given that AC is on the perpendicular bisector of BD, and that BD lies on the perpendicular bisector of AC. Prove that quadrilateral ABCD is a rhombus (a rhombus is a parallelogram whose four sides have the same length).

It is easy to prove that the four sides are of equal length, but I don't know how to prove that the pairs of sides are parallel. I think I should prove that the interior angles are supplementary, but I'm not sure how to do that. Any tips?
 

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May 2010
16
6
Sydney - Australia
one way to do this could be by first siding the sides by Cosine Rule (or Pythagoras) then applying sine rule in one of the traingles to find an accute angle. Find the alternate angle on another traingle, and if they turn out to be equal, then if two angles are alternately equal, then the lines must be parallel.

hope that helps :)

-sugarT

EDIT: easier would be just to do Cosine Rule to find the alternate angles.
 
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Apr 2010
118
6
Thank you so much for your suggestion! I don't think the book covers the sine/cosine rules though. Is there a simpler way to prove it? It seems like I've done a similar thing before, but I'm blanking out for some reason.
 
May 2010
16
6
Sydney - Australia
My pleasure. A simpler method could be by using triangular congruency, two sides and an angle in between. All triangles seem to have corresponding equal sides and an equal angle in between the two sides. Thus, they are congruent and hence we can find alternate equal angles which are in fact corresponding angles of the congruent triangles . Then from that say they must be parallel.
 
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May 2010
1
1
India
Lets consider ABCD to be a quadrilateral
AC intersects BD perpendicularly at say O
AOD = COB = 90
BO is common
ABO = CBO since when BD intersects AC perpendicularly
hence triangle ABO=triange BOC
hence AB=CB and AO = OC
Similarly it can be proved that triangle ADO = triangle DOC
hence AD=DC
similarly, DC = BC and BO = OD, AB = AD
hence proved that all the sides are equal
Now considering the triangles AOD and triangle BOC,

AD =BC, AOD=BOC and BO=OD hence the triangles are equal.
Hence , ADO = CBO and DAO = BCO, proving that AD||BC
Similarly it can be proved for AB and CD
 
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Apr 2010
118
6
Thank you both! I was looking at the wrong sets of lines.