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Mathematics

Area of a Triangle Given Its Vertices

am1m2, am1+m2...

Question

${a m_{1} m_{2}, a (m_{1} + m_{2})}, {a m_{2} m_{3}, a (m_{2} + m_{3})} a n d {a m_{3} m_{1}, a (m_{3} + m_{1})}$ .

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Solution

Area of triangle having angular points
$(x_{1}, y_{1})$ $(x_{2}, y_{2})$ $(x_{3}, y_{3})$ is given by the formula

$△ = \frac{1}{2} | x_{1} y_{2} + x_{2} y_{3} + x_{3} y_{1} - x_{2} y_{1} - x_{3} y_{2} - x_{1} y_{3} |$
or $△ = \frac{1}{2} | (x_{1} - x_{3}) y_{2} + (x_{2} - x_{1}) y_{3} + (x_{3} - x_{2}) y_{1} |$
$△ = \frac{1}{2} | (a m_{1} m_{2} - a m_{3} m_{1}) a (m_{2} + m_{3}) + (a m_{2} m_{3} - a m_{1} m_{2}) a (m_{3} + m_{1}) + (a m_{3} m_{1} - a m_{2} m_{3}) a (m_{1} + m_{2}) |$
$\Rightarrow △ = \frac{1}{2} a^{2} | m_{1} (m_{2} - m_{3}) (m_{2} + m_{3}) + m_{2} (m_{3} - m_{1}) (m_{3} + m_{1}) + m_{3} (m_{1} - m_{2}) (m_{1} + m_{2}) |$

$\Rightarrow △ = \frac{1}{2} a^{2} ∣ ∣ m_{1} (m_{2}^{2} - m_{3}^{2}) + m_{2} (m_{3}^{2} - m_{1}^{2}) + m_{3} (m_{1}^{2} - m_{2}^{2}) ∣ ∣$

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{am1m2,a(m1+m2)},{am2m3,a(m2+m3)}and{am3m1,a(m3+m1)}.

${a m_{1} m_{2}, a (m_{1} + m_{2})}, {a m_{2} m_{3}, a (m_{2} + m_{3})} a n d {a m_{3} m_{1}, a (m_{3} + m_{1})}$ .