Question 938786
an equilateral triangle is a triangle in which {{{all}}} {{{three}}} {{{sides}}} are {{{equal}}}

if a triangle  {{{XYZ}}} has vertices 

{{{X}}} at ({{{1}}},{{{1}}}), 
{{{y}}} at ({{{3}}},{{{1}}}) and 
{{{Z}}} at ({{{2}}},{{{4}}})

to show that the triangle is an equilateral triangle, show that the distance between each two points is same 

use distance formula to find the length of the all three sides


{{{XY=sqrt((3-1)^2+(1-1)^2) }}}

{{{XY=sqrt(2^2+0) }}}

{{{XY=sqrt(4)}}}

{{{XY=2}}}


{{{YZ=sqrt((2-3)^2+(4-1)^2)}}}
{{{YZ=sqrt((-1)^2+(3)^2)}}}
{{{YZ=sqrt(1+9)}}}
{{{YZ=sqrt(10)}}}
{{{YZ=3.162277660168379}}}



{{{XZ=sqrt((2-1)^2+(4-1)^2)}}}
{{{XZ=sqrt(1+9)}}}
{{{XZ=sqrt(10)}}}
{{{XZ=3.162277660168379}}}



As {{{YZ}}} and {{{XZ }}} are equal,the triangle is an {{{isosceles}}} triangle , and (by definition) an isosceles triangle is a triangle with (at {{{least}}}) two equal sides 
 
an {{{equilateral}}} triangle is a triangle with with all sides equal 
an {{{equilateral}}}  triangle is therefore a special case of an {{{isosceles}}} triangle having not just two, but all three sides and angles equal