Two lines and a point

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Two parallel lines, $m$ and $n$ , and a point $P$ , are given, where $P$ does not lie between the lines. The distance from $P$ to $m$ is double the distance from $P$ to $n$ . How many uses of a straightedge and compass does it take to draw a line through $P$ perpendicular to $m$ and $n$ ?

All terminology in this question is explained in the first note of my straightedge and compass set. More straightedge and compass constructions can be found there.

3 2 4 5

1 solution

Sharky Kesa
Sep 1, 2016

Since the distance is double between each line, it means that that the distance between $n$ and $P$ is equal to the distance between $n$ and $m$ . Thus, if we pick an arbitrary point on $n$ and draw a circle centred at it passing though $P$ , it will intersect $m$ at $P'$ such that their distances from $n$ are the same by symmetr. This implies $PP'$ is perpendicular to $m$ and $n$ so only 2 constructions we required.

Two lines and a point

1 solution

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