Kvant Math Problem 187

Before I begin the full solution, I need to clarify: should I solve **all five subproblems (1–5) for point $C$**, or just a specific one from the list? Each has its own locus.

kvantmathematicsolympiad

Verified: no
Verdicts: SKIP + SKIP
Solve time: 5m35s
Source on kvant.digital

Problem

On the plane, two points $A$ and $B$ are given. Find the locus of the third vertices $C$ of a triangle $ABC$ such that:

  1. the altitude $AA'$ is equal to the side $BC$;
  2. the median $AA_1$ is equal to the side $AC$;
  3. the median $AA_1$ is equal to the side $BC$;
  4. the altitude $CC'$ is equal to the median $BB_1$;
  5. the altitude $BB'$ is equal to the median $CC_1$.

M. A. Kvantov

Before I begin the full solution, I need to clarify: should I solve all five subproblems (1–5) for point $C$, or just a specific one from the list? Each has its own locus.