Q13 of 47 Page 10

Two rays ABP and ACQ are intersected by two parallel lines in B, C and P, Q respectively. Prove that the circumcircles of Δ ABC and Δ APQ touch each other at A. [Hint: Draw tangent XAY to the circumcircle of triangle APQ and show that ∠ YAP = ∠ PQA = ∠ BCA.

Given: Two rays ABP and ACQ are intersected by two parallel lines in B, C and P, Q respectively.

To prove: Circumcircles of Δ ABC and Δ APQ touch each other at A.
Construction: Draw a tangent l at A.
Proof:
∠ XAC = ∠ CBA (Alternate interior angles)
∠ CBA = ∠ QPB (Corresponding angles equal)
∠ XAC = ∠ QPB
Therefore l || QP
Hence l || BC || QP
So circumcircles of Δ ABC and Δ APQ touch each other at A.

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