Answer
The lengths are $|PQ|=3$, $|PR|=6$, and $|QR|=3\sqrt{5}.$
The triangle is not isosceles but is a right triangle.
Work Step by Step
To find the lengths of the legs of the triangle, we must find the distance between each pair of points.
$$|PQ|=\sqrt{(4-2)^2+(1+1)^2+(1-0)^2}
\\=\sqrt{(2)^2+(2)^2+(1)^2} \\=\sqrt{4+4+1}\\=\sqrt{9}\\=3.$$
$$|PR|=\sqrt{(4-2)^2+(-5+1)^2+(4-0)^2}
\\=\sqrt{(2)^2+(-4)^2+(4)^2} \\=\sqrt{4+16+16}\\=\sqrt{36}\\=6.$$
$$|QR|=\sqrt{(4-4)^2+(-5-1)^2+(4-1)^2}
\\=\sqrt{(0)^2+(-6)^2+(3)^2} \\=\sqrt{0+36+9}\\=\sqrt{45}\\=\sqrt{9(5)}\\=3\sqrt{5}.$$
Now, if a triangle has two sides of equal length, then it is isosceles. So since the sides of our triangle all have different lengths, our triangle is not isosceles.
And for a triangle to be a right triangle, the square of the longest side must equal the sum of the squares of the other two sides. The longest side of our triangle is $|QR|=3\sqrt{5}$, and $$|PQ|^2+|PR|^2=9+36=45=(9\sqrt{5})^2=|QR|^2.$$
So our triangle is a right triangle.
