Question

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prove that the quadrilateral formed (if possible) by the external angle bisectors AH, BF, CF and DH of internal angles A, B,C and D respectively form a quadrilateral EFGH. ​

Answer 1

Answer:

Hey Mate ,

→ Given question :  prove that the quadrilateral formed (if possible) by the external angle bisectors AH, BF, CF and DH of internal angles A, B,C and D respectively form a quadrilateral EFGH. ​

Step-by-step explanation:

Solution :

→ ∠ FEH = ∠ AED = 180°  - ∠ EAB - ∠ EBA

→                            =  180° -  1 /2 ( ∠A + ∠ B )

→ and ∠ FGH = ∠ CGD = 180° - ∠ GCD - ∠ GDC

→                                     =  180° - 1/2 ( ∠ C + ∠ D )

→  Therefore , ∠ FEH + ∠ FGH = 180° - 1 / 2 ( ∠A + ∠B) + 180° - 1 /2 ( ∠C + ∠D)

→                                                =   360° - 1/2 (  ∠ A + ∠B + ∠C + ∠D ) = 360° -

→                                                =  1 / 2 × 360°

→                                                = 360° - 180° = 180°

→  Therefore ,  the quadrilateral EFGH is cyclic.

→ Thank you

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