Overview
A geodesic in a metric space can be thought of as an isometry map from \([0,1]\) to the space. We call a space as geodesic metric space if there exists a geodesic between any two points. Consider a geodesic metric space \(X\) that admits a bounded combing.
- Reflection about line \(BC\).
- Reflection about line \(CA\).
- Reflection about line \(AB\).
The goal is to determine if any point \(P\) returns to its original position after these three reflections.
Analysis
Let \(A'\) be the reflection of \(A\) over \(BC\), and \(C'\) be the reflection of \(C\) over \(AB\). Suppose \(P\) is a point that maps to itself after the sequence. After the reflections:
- \(A' \curvearrowright A\) (where \(\curvearrowright\) denotes "maps to"),
- \(C \curvearrowright C'\).
Since reflection is an isometry, it preserves distances and angles. Thus, we have: \[ \triangle PA'C \cong \triangle PAC' \] This implies \(P\) is the center of a spiral similarity mapping \(\overline{A'C}\) to \(\overline{AC'}\). Such a point, if it exists, is unique.
Conclusion
Further analysis shows that for \(\triangle BA'C \cong \triangle BAC'\):
- \(BA' = BA\),
- \(A'C = AC = AC'\),
- \(BC = BC'\).
This suggests \(B\) could be a candidate for \(P\). However, testing reveals \(B\) moves with each reflection and does not return to itself, regardless of the triangle’s shape. Thus, no fixed point \(P\) exists in the plane under this transformation sequence.
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