Choose whether the given statement is true or false.

If a hexagon ABCDEF circumscribes a circle, then it is true that AB + CD + EF = BC + DE + FA.

Subtopics: Determinant of matrices

Question

Choose whether the given statement is true or false.If a hexagon ABCDEF circumscribes a circle, then it is true that AB + CD + EF = BC + DE + FA.Subtopics: Determinant of matrices

Accepted Answer

We are given a hexagon ABCDEF that circumscribes a circle, we know circumscribing a circle means the hexagon contains a circle which touches all its edges at some point. Let circle touch hexagon at L, M, N, O, P and Q. So,
Question Image

Now we know, a pair of tangents coming from the same external point to a circle are equal in length.
Now, for the circle AQ and AL are tangent that is from same point A. So,
AQ=AL⋯⋯⋯⋯⋯⋯(1)

Similarly, FQ and FP are the tangent to the circle from the same external point F. So,
FQ=FP⋯⋯⋯⋯⋯⋯(2)

Now, also EP and EO are tangent from same external point E. So,
EO=EP⋯⋯⋯⋯⋯⋯(3)
Also, we have DO and DN are tangent to circle from external same point D. So,
DO=DN⋯⋯⋯⋯⋯⋯(4)

Now, NC and MC are tangent to circle from same external point C. So,
MC=NC⋯⋯⋯⋯⋯⋯(5)

Lastly, BL and BM are tangent to circle from same external point B. So,
BM=BL⋯⋯⋯⋯⋯⋯(6)

Now, we are asked to show AB+CD+EF=BC+DE+FA

We consider right hand side BC+DE+FA we can see from figure, we can split side of hexagon as
BC+DE+FA=(BM+MC)+(DO+EO)+(FQ+QA)

Now using (1), (2), (3), (4), (5) and (6) we get:
⇒ (BL+NC)+(DN+EP)+(AL+FP)
Now, we arrange the terms, so we get:
⇒ (BL+AL)+(NC+DN)+(EP+FP)
Now, (BL+AL=AB),(NC+DN=CD) and (EP+FP=EF) so we get:
⇒ AB+CD+EF.
Hence, the given statement is true.

Choose whether the given statement is true or false.

If a hexagon ABCDEF circumscribes a circle, then it is true that AB + CD + EF = BC + DE + FA.

Subtopics: Determinant of matrices

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