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| Concurrency of the perpendicular bisectors of the sides of a triangle |
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| 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). |
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| Recall the procedure to draw the perpendicular bisectors of the sides of a triangle. |
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 Draw DABC. |
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 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). |
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 Similarly, draw the perpendicular bisectors of AB and AC. |
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 The three perpendicular bisectors of the sides of DABC pass through a common point (S). |
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| 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. |
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| 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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