Condition for Points in Complex Plane to form Isosceles Triangle/Examples/1+2i, 4-2i, 1-6i

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Examples of Use of Condition for Points in Complex Plane to form Isosceles Triangle

Let $A = z_1 = 1 + 2 i$, $B = z_2 = 4 - 2 i$ and $C = z_3 = 1 - 6 i$ represent on the complex plane the vertices of a triangle.


Then $\triangle ABC$ is isosceles, where $B$ is the apex.


Proof

By Condition for Points in Complex Plane to form Isosceles Triangle:

$\triangle ABC$ is isosceles, where $A$ is the apex, if and only if $AB = AC$.


Hence:

\(\ds AB\) \(=\) \(\ds \cmod {z_1 - z_2}\)
\(\ds \) \(=\) \(\ds \cmod {\paren {1 + 2 i} - \paren {4 - 2 i} }\)
\(\ds \) \(=\) \(\ds \cmod {\paren {1 - 4} + \paren {2 - \paren {-2} i} }\)
\(\ds \) \(=\) \(\ds \cmod {-3 + 4 i}\)
\(\ds \) \(=\) \(\ds \sqrt {3^2 + 4^2}\)
\(\ds \) \(=\) \(\ds 5\)


Similarly:

\(\ds BC\) \(=\) \(\ds \cmod {z_2 - z_3}\)
\(\ds \) \(=\) \(\ds \cmod {\paren {4 - 2 i} - \paren {1 - 6 i} }\)
\(\ds \) \(=\) \(\ds \cmod {\paren {4 - 1} + \paren {-2 - \paren {-6} i} }\)
\(\ds \) \(=\) \(\ds \cmod {3 + 4 i}\)
\(\ds \) \(=\) \(\ds \sqrt {3^2 + 4^2}\)
\(\ds \) \(=\) \(\ds 5\)


So $AB = BC$ and so $\triangle ABC$ is isosceles.


Finally note that:

\(\ds AC\) \(=\) \(\ds \cmod {z_1 - z_3}\)
\(\ds \) \(=\) \(\ds \cmod {\paren {1 + 2 i} - \paren {1 - 6 i} }\)
\(\ds \) \(=\) \(\ds \cmod {\paren {1 - 1} + \paren {2 - \paren {-6} i} }\)
\(\ds \) \(=\) \(\ds \cmod {8 i}\)
\(\ds \) \(=\) \(\ds 8\)


demonstrating that $\triangle ABC$ is not equilateral.

$\blacksquare$


Sources

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