Concurrent Lines


   
 
Concurrency of the perpendicular bisectors of the sides of a triangle
 
Draw a triangle ABC. Draw the perpendicular bisectors of its sides. The perpendicular bisectors of the sides of a triangle are concurrent (pass through the same point).
 
Recall the procedure to draw the perpendicular bisectors of the sides of a triangle.
 
Draw DABC.
 
With B and C as centres and radius equal to more than half of BC draw arcs of circles on either side of BC as to intersect at X and Y. Join XY (extend the bisecting line if needed).
 
Similarly, draw the perpendicular bisectors of AB and AC.
 
The three perpendicular bisectors of the sides of DABC pass through a common point (S).
 
Theorems on concurrence
 
Through the activities suggested you will find that the angle bisectors, the perpendicular bisectors of sides, the altitudes and the medians of a triangle are concurrent. Now medians of a triangle are concurrent. Now we shall prove these properties by logical deductions. The point of concurrency are stated as theorems and proved.
 
Note:
In proving theorems on concurrence, we draw two lines which intersect at a point and draw the third line in such a way that it passes through the point of intersection and then we prove that this straight line obeys the given conditions.
 
 
     
   
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