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Constructing Minimal Area (Posted on 2010-05-13) Difficulty: 3 of 5
In Minimal area, we were asked to find the property of a line which minimized the area of the triangle it completed. It turned out the given point was the midpoint of the third side. In this problem, I am asking for a ruler and compass construction of that line, given the original angle and point.

See The Solution Submitted by Brian Smith    
Rating: 3.0000 (1 votes)

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Solution Solution | Comment 1 of 5

See my solution for the "Minimal area"
problem for notation and solution.
If a = b, then construct the perpendicular
to the line VW through W.
If a = 90, then construct the perpendicular
to the line VW through W intersecting one
of the original rays at point B. Construct
point X on the other ray such that
|VX| = 2|WB|. Line XW is the desired line.
If a <> 90, then construct the perpendicular
to the line VW through W intersecting the
original rays at points A and B ( where
a = angle WVA). Construct
the circle passing through A, B, and V
intersecting ray VW again at U. Construct
point U' on line segment WB such that
|WU'| = |WU|. Construct point S on line
segment WA such that |AS| = |WB|. Construct
point T as the midpoint of line segment WS.
Construct point T' on line segment WV such
that |WT'| = |WT|. Construct the desired
line XY through W and parallel to line T'U'.
Proof of this construction.
                          |WU'|     |WU|
 tan(VWX) = tan(WT'U') = ------- = ------
                          |WT'|     |WT|
              |WA||WB|       2|WA||WB|
             ----------     -----------
                |WV|            |WV|
          = ------------ = -------------
               |WS|/2       |WA| - |WB|
                |WA|   |WB|
             2 ------*------
                |WV|   |WV|
          = -----------------
              |WA|     |WB|
             ------ - ------
              |WV|     |WV|
             2 tan(WVX)*tan(WVY) 
          = ---------------------.
             tan(WVX) - tan(WVY)
 

Edited on May 14, 2010, 6:09 pm
  Posted by Bractals on 2010-05-13 12:55:53

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