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Hexagon Circle
Here ABCDEF is a hexagon which circumscribe a circle.

Since the tangents drawn from a point to the circle are equal in length.

therefore AM = AR,

BM = BN,

CN = CO,

DO = DP,

EP = EQ,

FQ = FR.

now from these equations we have,

AM + BM = AR + BN⇒AB = AR + BN

CO + DO = CN + DP⇒CD = CN + DP

EQ + FQ = EP + FR⇒EF = EP + FR

now on addition these three equation gives

AB + CD + EF = AR + (BN + CN) + (DP + EP) + FR = BC + DE + FA

hence proved.

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