# P,Q and R are the midpoints of AO,BO and CO respectively as shown in figure.Prove that trangle ABC and triangle PQR are equiangular

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P is the midpoint of OA and Q is the midpoint of OB

By midpoint theorem,

PQ║AB

⇒∠OPQ=∠OAB(alt. opp angles)⇒1

Similarly

∠OPR=∠OAC⇒2

∠ORP=∠OCA⇒3

∠ORQ=∠OCB⇒4

∠OQR=∠OBC⇒5

∠OQP=∠OBA⇒6

By adding equations 1& 2 we get

∠OPQ+∠OPR=∠OAB+∠OAC

⇒∠P=∠A

Similarly by adding 3&4 and 5&6 we get ∠C=∠R and∠B=∠Q respectively

Therefore ΔABC and ΔPQR are equiangular

Hence proved