Constructing a Triangle

Date: 09/29/2003 at 00:12:22
From: Bong 
Subject: constructing a triangle  

Let x be a given angle, and let and m and n be given lengths such that
n > m. How can I construct triangle ABC such that AB = m, AC + CB = n,
and the measure of angle ACB = x?

I tried to add n and m and use the sum as the perimeter of the
triangle but I can't construct a triangle with given segment AB and
angle x.

Date: 10/01/2003 at 03:23:00
From: Doctor Floor
Subject: Re: constructing a triangle  

Hi, Bong,

We will use several times that we can construct the circle containing 
the points that make a given angle y on a segment UV.  This can be
done in the following way:

The circle we look for passes through U and V.  We need a third point.

From V draw the perpendicular line to UV.  From U draw the line making 
an angle of 90-y with UV.  These lines intersect in a point W.  Angle 
UWV is then equal to y.  So W lies on the required circle. 

That means that the circumcircle of UVW is the required circle.  Since
UVW is a right triangle, this circle has the midpoint of the
hypotenuse UW as center.

Now let's analyze the problem you have given.  If we consider a
triangle ABC with angle A = x, AB = m, BC = n1 and CA = n2, then we
consider the point D on BC produced such that CD = n2, or BD = n1 + n2
= n.

In the following diagram,

  

triangle ACD is isosceles, and from angle ACD = 180 - x we see that
angle ADB = x/2. Of course, with x given, we can construct an angle of
x/2 by constructing the angle bisector.

This gives the following construction:

  Start with AB of given length.

  Let C1 be the circle with center B and radius n. Let C2 be 
  the circle of points making angle x/2 on AB. The points of
  intersection of C1 and C2 give possible D. C lies on BD. 

  Construct the circle C3 of points making angle x on AB. The 
  intersection apart from B of BD and C3 gives C.

I wish to thank Antreas P. Hatzipolakis for suggesting to me this 
construction.

If you have more questions, just write back.

Best regards,

- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/06/2003 at 03:22:38
From: Bong Alvarez
Subject: Thank you (constructing a triangle)

I wish to thank you for answering my question.  It was a big help.

I was wondering if this construction can be done using a straight edge
and compass?

Date: 10/06/2003 at 06:17:54
From: Doctor Floor
Subject: Re: Thank you (constructing a triangle)

Hi, Bong,

Indeed it can.  Perhaps you need some basic constructions, such as the 
ones given in the Dr. Math library at:

  http://mathforum.org/library/drmath/view/55076.html 
  http://mathforum.org/library/drmath/view/54688.html 

If you have more questions, just write back.

Best regards,

- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/