Prove that the quadrilateral formed ( if possible) by the internal angle bisector of any quadrilateral is cyclic.
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Prove that the quadrilateral formed ( if possible) by the internal angle bisector of any quadrilateral is cyclic.
no copied answer needed
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→ Given Question:-
→ Prove that the quadrilateral formed ( if possible) by the internal angle bisector of any quadrilateral is cyclic.
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→ Solution:-
⇒ In The Given fig, PSRQ is a Quadrilateral ,
⇒ Now , the angle bisectors are PH, SF,RF, and QH of internal angles P, S,R, and Q respectively form a quadrilateral EFGH .
⇒ Now, ∠FEH = ∠PES
⇒ = 180°-∠EPS - ∠ESP
⇒ 180° - [tex]\frac{1}{2}[/tex] ( ∠P + ∠ S )
⇒ and ∠ FGH = ∠ RGQ
⇒ = 180° - ∠ GRQ - ∠ GQR
⇒ 180° - [tex]\frac{1}{2}[/tex] ( ∠R + ∠Q )
⇒ ∴ ∠FEH + ∠ FGH
⇒ 180 - [tex]\frac{1}{2}[/tex] ( ∠P + ∠S) + 180° -
⇒ 360° - [tex]\frac{1}{2}[/tex] ( ∠P + ∠S + ∠R + ∠Q )
⇒ 360° -[tex]\frac{1}{2}[/tex] × 360° = 180°
⇒ Hence , EFGH is a cyclic quadrilateral.
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