Question

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

Answer 1

In ΔAOB,
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