# Van Schooten's Ruler Constructions - Solution to Problem IV

Author(s):
C. Edward Sandifer

Problem IV:   Above a given indefinitely long straight line, to construct a perpendicular.

Construction:  Conceive the given straight line as going through points A and B, and a perpendicular is to be constructed above it; make BC equal to AB [along the same line] and from B draw BD, making with AB any angle whatsoever, and locate D on that line so that it equals BA or BC, and draw the line from point D through point C.  If in that line CF is made equal to CA, and in the line ABC, CE is made equal to CD, I say joining EF makes it be perpendicular to AB.

This may be van Schooten’s trickiest construction.  The key to the proof of correctness is to note that the points A, C and D are all the same distance from B, so they lie on a circle centered at B and with AC as a diameter.  This makes angle ADC a right angle.  Now you only have to figure out why angle CEF is also a right angle.

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Solution to Problem V

The other eight problems

C. Edward Sandifer, "Van Schooten's Ruler Constructions - Solution to Problem IV," Loci (August 2010)