Answer
A right isosceles triangle
(both).
Work Step by Step
$d(P_{1},P_{2})=\sqrt{(6-(-1))^{2}+(2-4)^{2}}$
$=\sqrt{7^{2}+(-2)^{2}}=\sqrt{49+4}=\sqrt{53}$
$d(P_{1},P_{3})=\sqrt{(4-(-1))^{2}+(-5-4)^{2}}$
$=\sqrt{5^{2}+(-9)^{2}}=\sqrt{25+81}=\sqrt{106}$
$d(P_{2},P_{3})=\sqrt{(4-6)^{2}+(-5-2)^{2}}$
$=\sqrt{(-2)^{2}+(-7)^{2}}=\sqrt{4+49}=\sqrt{53}$
Two sides have the same legth - isosceles triangle.
To check whether it is a right triangle,
the longest side is $c=\sqrt{106}$ . Check whether $a^{2}+b^{2}=c^{2}$
$a^{2}+b^{2}=(\sqrt{53})^{2}+(\sqrt{53})^{2}=53+53=106=c^{2}$.
A right isosceles triangle
(both).