Tons of Triangles

All points on a Cartesian plane are colored either $\color{#D61F06}{\text{red}}$ or $\color{#3D99F6}{\text{blue}}$.

For all positive numbers $r,$ there exists a non-degenerate isosceles triangle with legs of length $r$ whose vertices are all colored $\text{\_\_\_\_\_\_\_\_\_\_}$.

Is the statement above true for at least one of the colors?

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Note: A degenerate triangle is one where all three vertices lie on a straight line.