In the given figure, ABC is an equilateral triangle, PQ ‖AC and AC is produced to R such that CR=BP. Prove that QR bisects PC.
Given: ABC is an equilateral triangle, PQ ‖AC and CR=BP
To prove: QR bisects PC or PM = MC
Proof:
Since, ∆ABC is equilateral triangle,
∠A = ∠ACB = 60°
Since, PQ ‖AC and corresponding angles are equal,
∠BPQ = ∠ACB = 60°
In ∆BPQ,
∠B= ∠ACB = 60°
∠BPQ = 60°
Hence, ∆BPQ is an equilateral triangle.
∴ PQ = BP = BQ
Since we have BP = CR,
We say that PQ = CR …(1)
Consider the triangles ∆PMQ and ∆CMR,
∠PQM = ∠CRM …alternate angles
∠PMQ = ∠CMR … vertically opposite angles
PQ = CR … from 1
Thus by AAS property of congruence,
∆PMQ ≅ ∆CMR
Hence, we know that, corresponding parts of the congruent triangles are equal
∴ PM = MC