Understand the problem
Source of the problem
Start with hints
From the last hint, we have where is the circumradius of . Hence, is a rhombus and . Similarly, hence . As is a parallelogram, we also have . We can similarly prove that and . Thus . This implies that it suffices to show that the centres of the two nine-point circles coincide. Remember that the nine-point centre is the midpoint of the line joining the circumcentre and the orthocentre. Claim is the circumcentre of . Proof Note that (as is the centre of and is a reflection of . Similarly, . Claim is the orthocentre of . Proof As is the reflection of on , . As , . Similarly, and . Thus is the orthocentre of .
Thus the centres of the nine-point circles of and coincide.
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