Problem 5

Suppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maximum number of intersections that these perpendiculars can have.

Step 2 of 5: Bound intersections on one line

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20\text{ intersections on each chosen perpendicular}

Detailed analysis

Fix the perpendicular from $A$ to $BC$ . The source count gives at most $5$ intersections from perpendiculars based at $B$ and $C$ each, and at most $5$ from each of $D$ and $E$ (the perpendiculars to $BC$ are parallel and do not meet it). Thus it meets at most $20$ other relevant perpendiculars, apart from forced vertex and orthocenter coincidences.