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 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 = 2WB. 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
WAWB 2WAWB
 
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 20100513 12:55:53 