# Understand the problem

##### Source of the problem

##### Topic

##### Difficulty Level

##### Suggested Book

# 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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Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

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