Question

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

Answer 1

Let ABCDEF be a hexagon.It is circumscribe a circle.

AM = AR -----(i)
BM = BN ---------(ii)
CN = CO ----------(iii)
DO = DP ----------(iv)
EP = EQ ----------(v)
FQ = FR --------- (vi)
adding (i) and (ii) we get
AM + BM = AR + BN
⇒ AB = AR + BN
adding (iii) and (iv) we get
CO + DO = CN + DP
⇒ CD = CN + DP
adding (v) and (vi) we get
EQ + FQ = EP + FR
⇒ EF = EP + FR
Adding all these we obtain
AB + CD + EF = AR + ( BN + CN ) + (DP + EP) + FR = BC + DE + FA
∴ AB + CD + EF = BC + DE + FA