Solver To determine if 3 points lie in a line

Source code of 'To determine if 3 points lie in a line'

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==section input
Enter the coordinates of 3 points lying on a plane:

X1= *[input x1=1]       Y1= *[input y1=1]
X2=*[input x2=2]        Y2=*[input y2=2]
X3=*[input x3=3]        Y3=*[input y3=3]

To find if these points lies on a line.

==section solution
perl
my $slope=(($y2-$y1)/($x2-$x1));
my $slope1=(($y3-$y1)/($x3-$x1));
my $slope2=(($y3-$y2)/($x3-$x2));

if(($slope==$slope1) && ($slope==$slope2))
{
print " The 3 points lie on a same plane. For all points to lie on a line they should satisfy the <A HREF=http://www.algebra.com/algebra/homework/Linear-equations/Polynomial-Expressions-and-Equations-in-Color.lesson>equation of a line</A>. Hence any two points taken on a line should calculate to the same slope of a line.

In order to prove the 3 points to lie on a line, as there exists a unique line containing three points and every line has a unique <A HREF=http://www.algebra.com/algebra/homework/Graphs/slope.wikipedia>slope</A>. 
Hence it will be sufficient to prove that the slope calculated taking 2 points at a time should be equal.

Slope of line taking points (X1,Y1) and (X2,Y2) is

{{{slope = (Y2-Y1)/(X2-X1)}}}

{{{slope = (($y2-$y1)/($x2-$x1)) = $slope}}}     ........................(1)

Slope of line taking points (X3,Y3) and (X1,Y1) is

{{{slope = (Y3-Y1)/(X3-X1)}}}

{{{slope = (($y3-$y1)/($x3-$x1)) = $slope1}}}    ........................(2)

From conditions (1) and (2)

The slopes are equal hence the 3 points can lie on same line.

If the slope calculated from points (X2,Y2) and (X3,Y3) comes out to be same then it is confirmed that the 3 points lie on a same line.

{{{slope = (Y3-Y2)/(X3-X2)}}}

{{{slope = (($y3-$y2)/($x3-$x2)) = $slope2}}}    ........................(3)

From (1),(2) and (3)

Hence, It is proved that the 3 points lie on same line.

To read more on equations of a line refer to articles on <A HREF=http://www.algebra.com/algebra/homework/Graphs/Graphing-by-Slope-Intercept-Method.lesson>wikipedia</A>
"
}

else
{ 
print " The 3 points lie on a same plane. For all points to lie on a line they should satisfy the <A HREF=http://www.algebra.com/algebra/homework/Linear-equations/Polynomial-Expressions-and-Equations-in-Color.lesson>equation of a line</A>. Hence any two points taken on a line should calculate to the same slope of a line.

Hence it will be sufficient to prove that the slope calculated taking 2 points at a time should be equal.

Slope of line taking points (X1,Y1) and (X2,Y2) is

{{{slope = (Y2-Y1)/(X2-X1)}}}

{{{slope = (($y2-$y1)/($x2-$x1)) = $slope}}}     ........................(1)

Slope of line taking points (X3,Y3) and (X1,Y1) is

{{{slope = (Y3-Y1)/(X3-X1)}}}

{{{slope = (($y3-$y1)/($x3-$x1)) = $slope1}}}    ........................(2)

From conditions (1) and (2)

The 3 points do not a same line.

For all points to lie on a line they should satisfy the equation of a line. Hence any two points taken on a line should calculate to the same slope of a line.

Here the slopes are unequal hence the points do not lie on same line.

To read more on equations of a line refer to articles on <A HREF=http://www.algebra.com/algebra/homework/Graphs/Graphing-by-Slope-Intercept-Method.lesson>wikipedia</A>

";
}

==section output
slope1,

==section check
  x1=6 y1=6 x2=2 y2=2 x3=0 y3=0 slope=1