Question 119
Initially, three non-collinear points, $A$, $B$, and $C$, are marked on the plane. You have a pencil and a double-edged ruler of width 1. Using them, you may perform the following operations:
- Mark an arbitrary point in the plane.
- Mark an arbitrary point on an already drawn line.
- If two points $P_1$ and $P_2$ are marked, draw the line connecting $P_1$ and $P_2$.
- If two non-parallel lines $\ell_1$ and $\ell_2$ are drawn, mark the intersection of $\ell_1$ and $\ell_2$.
- If a line $\ell$ is drawn, draw a line parallel to $\ell$ that is at distance 1 away from $\ell$ (note that two such lines may be drawn).
Prove that it is possible to mark the orthocenter of $A B C$ using these operations.
- Mark an arbitrary point in the plane.
- Mark an arbitrary point on an already drawn line.
- If two points $P_1$ and $P_2$ are marked, draw the line connecting $P_1$ and $P_2$.
- If two non-parallel lines $\ell_1$ and $\ell_2$ are drawn, mark the intersection of $\ell_1$ and $\ell_2$.
- If a line $\ell$ is drawn, draw a line parallel to $\ell$ that is at distance 1 away from $\ell$ (note that two such lines may be drawn).
Prove that it is possible to mark the orthocenter of $A B C$ using these operations.