Question

Let ABCDEF be an inscribed hexagon with AB=CD=EF and BC=DE=FA. Prove that $\vec{OA}+\vec{OB}+\vec{OC}+\vec{OD}+\vec{OE}+\vec{OF}=\vec{0}$/

Answer 1

Answer:

From the picture we have

AC

=

AB

+

BC

.....(1) and

AC

+

CD

=

AD

.....(2) and

AD

+

DE

=−

EA

....(3).

And

FA

=

EA

−

EF

......(4).

Now,

AB

+

AC

+

AD

+

EA

+

FA

=

AB

+(

AB

+

BC

)+(

AB

+

BC

+

CD

)+2

EA

−

EF

=

AB

+(

AB

+

BC

)+(

AB

+

BC

+

CD

)−2(

AB

+

BC

+

CD

+

DE

)−

EF

=

AB

−

CD

−2

DE

−

EF

=

AB

−

CD

+2

AB

+

CD

[Since

DE

=−

AB

and

FA

=−

CD

]

=3

AB

.

Comparing with the problem we have λ=3.

solution

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