is it always true that randomly given three line segments of equal length always forms an equilateral triangle.

Question

Is it possible to conclude, if three line segments are equal in length then they always form an equilateral triangle at their common intersection points?

Answer 1

The answer is: no.

Let in $\Delta ABC$ $AB=BC=13$, $AC=10$, $D$ be a midpoint of $AC$ and $E\in BC$ such that $AE=12$.

Let $BE=x$.

Thus, by law of cosines $$12^2=13^2+x^2-2\cdot x\cdot13\cdot\frac{2\cdot13^2-10^2}{2\cdot13^2}$$ and take a smallest root of this equation.

Now, let $F\in AB$ such that $BF=x$.

Thus, $AE=BD=CF$ and these segments have common point, but the $\Delta ABC$ is not equilateral triangle.

In another formulation it's also not necessary true.

Take not-equilateral triangle $ABC$ and take segments $MN=KL=FE$ such that $AB\subset MN$, $AC\subset KL$ and $BC\subset FE$.