Solver Calculate side length of equilateral triangle inscribed in the circle

Source code of 'Calculate side length of equilateral triangle inscribed in the circle'

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==section input
The radius of circle is *[input radius=6]. Find the sides of an equilateral triangle inscribed in it.

==section solution
perl
if($radius<=0)
{
print " Please enter the value greater than zero to calculate sides of equilateral inscribed in a circle.";
}
else{
my $side=1.7320508*$radius;
my $x=$radius/2;
my $y=$radius*1.7320508/2;
my $z=$radius+1;

print " 
The <A HREF=Circle.wikipedia>circle</A> of radius $radius with center O with an <A HREF=Triangles-and-its-basic-properties.lesson>equilateral triangle</A> ABC inscribed in it.

Join edges A,B and C with center O as shown in figure.

{{{drawing( 160, 160, -$z, $z, -$z, $z, locate(0.3,0.5,O),locate(0,($radius+1),C),locate(-($x+0.5),-($y+0.5),A),locate(($x+0.5),-($y+0.5),B),circle(0,0,$radius), line(0,0,0,$radius),line(0,0,-$x,-$y),line(0,0,$x,-$y),line(0,$radius,-$x,-$y),line(-$x,-$y,$x,-$y),line($x,-$y,0,$radius))}}}

Consider Triangle AOB and AOC,

1.> {{{AB=AC}}}   .............(Sides of Equilateral triangle are equal)

2.>{{{OC=OB}}}    .............(These are radius of a circle)

3.>OA is common side to both <A HREF=Triangles-and-its-basic-properties.lesson>Triangles</A>.

From conditions 1,2 and 3

Triangle AOB and AOC are <A HREF=http://www.algebra.com/algebra/homework/Triangles/Geometry-Proving-Triangles-Congruent.lesson>congruent</A> to each other. (SSS congruency condition)

Now since triangles are congruent,

Therefore, {{{Angle OAB=Angle OAC}}}    ...............(4)

Similarly, {{{Angle OBC= Angle OBA}}}   ...............(5)

As, <A HREF=Angle.wikipedia>Angle</A> CAB is an angle in a equilateral triangle

Hence, {{{Angle CAB=60}}} <A HREF=Degree_%2528angle%2529.wikipedia>degrees</A>       ...............(6)

Now, {{{Angle CAB=Angle OAB + Angle OAC}}}

{{{Angle CAB=Angle OAB + Angle OAB}}}    ..........(From (4))

{{{60=2 Angle OAB}}}

{{{Angle OAB = 60/2}}} degrees

{{{Angle OAB=Angle OAC=30}}} degrees     ...........(6)

Similarly, {{{Angle CBA=60}}}

and from condition 5, {{{Angle OBA= Angle OBC=30}}} degrees     ...........(7)

Now consider Triangle AOB,

Sum of angles in a Triangle is 180 degrees.

Hence, {{{Angle OAB + Angle OBA + Angle AOB = 180}}}

{{{Angle AOB + 30 + 30 = 180}}}   ........(From 6 and 7)

{{{Angle AOB = 180-(30+30)}}}

{{{Angle AOB = 180-60 = 120}}}       ...........(8)

In Triangle AOB using <A HREF=Sine_rule.wikipedia>sine rule</A>,

{{{(sin AOB/AB)=(sin OAB/OB) =(sin OBA/OA)}}}

{{{(sin 120/AB)=(sin 30/OB) =(sin 30/OA)}}}

{{{(sin 120/AB)=(sin 30/OB)}}}

{{{AB=(sin 120 * OB)/sin 30}}}

{{{AB=((sqrt(3)/2)*$radius)/(1/2)}}}

{{{AB=$side}}}

Hence the side of an equilateral triangle <A HREF=Inscribed_circle.wikipedia>inscribed</A>  in a circle of radius $radius is $side.

";
}

==section output
side,

==section check
  radius=40 side=69.282032