prove that the quadrilateral formed by bisectors of angles of any quadrilateral is a cyclic quadrilateral.
Answers
ABCD is a cyclic quadrilateral
∠A+∠C=180° and ∠B+∠D = 180°
(∠A+∠C)/2=90°and(∠B+∠D)/2= 90°
w,x,y,z are angles of the inner quadrilateral
x + z = 90°
y + w = 90°
In ΔAGD and ΔBEC,
x + y + ∠AGD = 180°
z + w + ∠BEC = 180°
∠AGD = 180° – (x+y)
∠BEC = 180° – (z+w)
∠AGD+∠BEC=360°-(x+y+z+w)
= 360° – (90+90) = 360° – 180° = 180°
∠AGD+∠BEC = 180°
∠FGH+∠HEF = 180°
The sum of a pair of opposite angles of a quadrilateral EFGH is 180°.
Hence EFGH is cyclic
Hence Proved
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Let assume that ABCD be a quadrilateral and let further assume that the quadrilateral formed by angle bisectors of angle A, angle B, angle C and angle D be PQRS.
Let assume that
Now, We know, sum of interior angles of a quadrilateral is 360°.
So, using this property, we have
Now, In triangle ARB
We know, sum of interior angles of a triangle is 180°.
So, using this property, we have
Now, In triangle CPD
We have
On adding equation (2) and (3), we get