A triangle has its three sides equal to a, b and c. If the coordinates of its vertices are A(x 1 y 1 ), B(x 2 , y 2 ) and C(x 3 y 3 ), show that - Wired Faculty

Question

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A triangle has its three sides equal to a, b and c. If the coordinates of its vertices are A(x₁ y₁), B(x₂, y₂) and C(x₃ y₃), show that
$space open vertical bar table row cell straight x subscript 1 end cell cell space space straight y subscript 1 end cell cell space 2 end cell row cell straight x subscript 2 end cell cell space straight y subscript 2 end cell cell space 2 end cell row cell straight x subscript 3 end cell cell space straight y subscript 3 end cell cell space 2 end cell end table close vertical bar squared space equals space left parenthesis straight a plus straight b plus straight c right parenthesis thin space left parenthesis straight b plus straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a minus straight b right parenthesis thin space left parenthesis straight a plus straight b minus straight c right parenthesis.$

Solution

We know that
$increment space equals space open vertical bar table row cell straight x subscript 1 end cell cell space straight y subscript 1 end cell 1 row cell straight x subscript 2 end cell cell straight y subscript 2 end cell 1 row cell straight x subscript 3 end cell cell straight y subscript 3 end cell 1 end table close vertical bar space space or space space space 2 increment space equals space open vertical bar table row cell straight x subscript 1 end cell cell space straight y subscript 1 end cell cell space 1 end cell row cell straight x blank presubscript 2 end cell cell space straight y subscript 2 end cell cell space 1 end cell row cell straight x subscript 3 end cell cell space straight y subscript 3 end cell cell space 1 end cell end table close vertical bar$
$rightwards double arrow space space 4 space increment space equals space space open vertical bar table row cell straight x subscript 1 end cell cell straight y subscript 1 end cell 2 row cell straight x subscript 2 end cell cell straight y subscript 2 end cell 2 row cell straight x subscript 3 end cell cell straight y subscript 3 end cell 2 end table close vertical bar space space or space space space space 16 increment squared space equals space open vertical bar table row cell straight x subscript 1 end cell cell space straight y subscript 1 end cell cell space 2 end cell row cell straight x subscript 2 end cell cell space straight y subscript 2 end cell cell space 2 end cell row cell straight x subscript 3 end cell cell space straight y subscript 3 end cell cell space 2 end cell end table close vertical bar squared space space space space space space space space space space space space... left parenthesis 1 right parenthesis$
Again, $increment space equals space square root of straight s left parenthesis straight s minus straight a right parenthesis thin space left parenthesis straight s minus straight b right parenthesis thin space left parenthesis straight s minus straight c right parenthesis end root space space where space straight s space equals space 1 half left parenthesis straight a plus straight b plus straight c right parenthesis$
$therefore space space space space space space space straight s minus straight a space equals space 1 half left parenthesis straight a plus straight b plus straight c right parenthesis minus straight a space equals space 1 half left parenthesis straight b plus straight c minus straight a right parenthesis$
Similarly, $straight s minus straight b space equals space 1 half left parenthesis straight c plus straight a minus straight b right parenthesis comma space space space straight s minus straight c space equals space 1 half left parenthesis straight a plus straight b minus straight c right parenthesis$
$therefore space space space space space space space space increment squared space equals space straight s left parenthesis straight s minus straight a right parenthesis thin space left parenthesis straight s minus straight b right parenthesis thin space left parenthesis straight s minus straight c right parenthesis rightwards double arrow space space space space space space space space increment squared space equals space 1 half left parenthesis straight a plus straight b plus straight c right parenthesis. space 1 half left parenthesis straight b plus straight c minus straight a right parenthesis thin space. space 1 half left parenthesis straight c plus straight a minus straight b right parenthesis. space 1 half left parenthesis straight a plus straight b minus straight c right parenthesis rightwards double arrow space space space space space 16 increment squared space equals space left parenthesis straight a plus straight b plus straight c right parenthesis space left parenthesis straight b plus straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a minus straight b right parenthesis thin space left parenthesis straight a plus straight b minus straight c right parenthesis rightwards double arrow space space space space space open vertical bar table row cell straight x subscript 1 end cell cell space space space straight y subscript 1 end cell cell space space 2 end cell row cell straight x subscript 2 end cell cell space space straight y subscript 2 end cell cell space space 2 end cell row cell straight x subscript 3 end cell cell space space straight y subscript 3 end cell cell space space 2 end cell end table close vertical bar squared space equals space left parenthesis straight a plus straight b plus straight c right parenthesis thin space left parenthesis straight b plus straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a minus straight b right parenthesis thin space left parenthesis straight a plus straight b minus straight c right parenthesis thin space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets$

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