The above picture is like the one in your link where XYO=<1, OYZ=<2, XZO=<4 and OZY=<3.
These 4 angles are equal because XY=XZ making the triangle isosceles and angles XYZ and XZY are equal.
The angle bisectors have been extended from O to meet XY and XZ at M and N.
Triangles MZY and NZY are congruent because angle MYZ=NZY (isosceles triangle) and <3=<2 and YZ is a common side. Therefore MY=NZ.
Triangles MOY and NOZ are congruent because <1=<4, angles MOY and NOZ are equal (opposite angles formed by intersecting straight lines) and MY=NZ. So we have one side and two angles congruent, therefore OY=OZ.