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Geometry: Cyclic Quadrilateral: Parallelogram
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Cyclic Quadrilateral: Parallelogram
A quadrilateral is called a cyclic quadrilateral if its all vertices lie on a circle. A cyclic quadrilateral has some special properties:
The sum of either pair of opposite angles of a cyclic quadrilateral is 180.
OR
The opposite angle of a cyclic quadrilateral are supplementary.
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Example: ABCD is a parallelogram. The circle through A, B and C intersects CD produced at E prove that AE = AD.
Solution: In order to prove that AE=AD i.e. AED is an isosceles triangle it is sufficient to prove that AED = ADE.
Since ABCE is a cyclic quadrilateral.
AED + ABC = 180 ...(i)
Now,
CDE is a straight line
ADE + ADC = 180
But, ADC and ABC are opposite angles of a parallelogram.
ADC = ABC
ABC + ADE = ADC + ADE
ABC + ADE = 180 ...(ii)
From equations (i) and (ii), you get
AED + ABC = ADE + ABC
AED = ADE
Thus, in A AED, you have AED = ADE
AD = AE.
 
   
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