Question

In triangle ABC, the medians BP and CQ are produced up to points M and N respectively such that BP = PM and CQ = QN. Prove that:

1. M, A, and N are collinear.
2. A is the mid-point of MN.

Answer 1

The figure is shown below

(i) In ΔAQN & ΔBQC 

AQ = QB (Given)

∠AQN = ∠BQC                       

QN = QC 

∴ ΔAQN ≅ ΔBQC                     ...[ by SAS  ] 

∴ ∠QAN = ∠QBC                   ...(1)

And BC = AN ……(2)

Similarly, ΔAPM ≅ ΔCPB           .....[by SAS] 

∠PAM = ∠PCB                      ...(3)  [by CPTC]        

And BC = AM                          ….( 4 )

Now In ΔABC,

∠ABC + ∠ACB + ∠BAC = 180°

∠QAN + ∠PAM + ∠BAC = 180°   ...[ (1), (2) we get ]

Therefore M, A, N are collinear.

(ii) From (3) and (4) MA = NA

Hence A is the midpoint of MN.