ABCD is a square. The bisector of ∠DBC cuts AC and CD at E and F respectively. Prove that BF × CE = BE × DF.
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Step-by-step explanation:
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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!
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