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.

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It is given that ABC is an equilateral triangle; PQ∥AC and AC is produced to R such that CR=BPConsider QR intersecting the line PC at point MWe know that △ABC is an equilateral triangleSo we get ∠A=∠ACB=60∘From the figure we know that PQ∥AC and ∠BPQ and ∠ACB are corresponding anglesSo we get∠BPQ=∠ACB=60∘Based on the △BPQ we know that∠B=∠ACB=60∘It can be written as∠BQP=60∘According to the figure we know that △BPQ is an equilateral triangleSo we getPQ=BP=BQIt is given that CR=BP so we getPQ=CR.(1)In the △PMQ and △CMR we know that PQ∥AC and QR is the transversalWe know that ∠PQM and ∠CRM are alternate angles and ∠PMQ and ∠CMR are vertically opposite angles∠PQM=∠CRM∠PMQ=∠CMRBy considering equation (1) and AAS congruence criterion△PMQ≅△CMRWe know that the corresponding parts of congruent triangles are equalPM=MCTherefore, it is proved that QR bisects PC.

Question Text	In the given figure, $A BC$ is an equilateral triangle; $PQ ∥ A C$ and $A C$ is produced to $R$ such that $CR = BP$ . Prove that $QR$ bisects $PC$ .
Answer Type	Text solution:1
Upvotes	150

In the given figure, $A BC$ is an equilateral triangle; $PQ ∥ A C$ and $A C$ is produced to $R$ such that $CR = BP$ . Prove that $QR$ bisects $PC$ .

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