Question

ABCD is a square. The bisector of ∠DBC cuts AC and CD at E and F respectively. Prove that BF × CE = BE × DF.

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

BF × CE = BE × DF

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

In figure, ☐ ABCD is a square.

∴ ∟A = ∟B = ∟C = ∟D = 90° - - [ By definition ]

Draw diagonals AC and BD. - - [ Construction ]

∴ m∠DBC = m∠BCA = m∠BDF = 45° - - ( 1 ) [ Diagonals of square bisect the opposite angles. ]

Also, draw the bisector of ∠DBC which cuts AC & CD at E and F respectively. - - [ Construction ]

∴ m∠DBF = m∠CBF - - ( 2 )

Now,

In ΔBDF & ΔBCE,

∠BDF ≅ ∠BCA - - [ From ( 1 ) ]

∠DBF ≅ ∠CBF - - [ From ( 2 ) ]

∴ ΔBDF ∼ ΔBCE - - [ AA test of similarity ]

→ BD / BC = DF / CE = BF / BE - - [ c.s.s.t. ]

→ DF / CE = BF / BE

→ DF × BE = BF × CE - - [ Cross multiplication ]

∴ BF × CE = BE × DF

Hence proved!