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  • If a hexagon ABCDEF circumscribe a circle, prove that AB + CD + EF = BC + DE + FA.

If a hexagon ABCDEF circumscribe a circle, prove that AB + CD + EF = BC + DE + FA.

Answers (1)

Solution
Given ABCDEF hexagon circumscribe a circle.

To Prove : AB + CD + EF = BC + DE + FA
Proof : Here AR = AS       [\because length of tangents drawn from a point are always equal]
Similarly,
BS = BT
CT = CU
DU = DV
EV = EW
FW = FR
Now AB + CD + EF = (AS + SB) + (CU + UD) + (EW + WF)
= (AR + BT) + (CT + DV) + (EV + FR)
= (AR + FR) + (BT + CT) + (DV + EV)
Now according to Euclid’s axiom when equals are added in equals then the result is also equal
\Rightarrow   AB + CD + EF = AF + BC + DE
i.e.,  AB + CD + EF = BC + DE + FA
Hence Proved

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