yes
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In the middle of the triangle.
No but a right angle triangle can be considered to be an isosceles triangle when its interior angles are 90, 45 and 45 degrees.
In the middle of the triangle
The median of a triangle cannot be considered as a base of that triangle.
Yes, every isosceles triangle has at least one line of symmetry, usually drawn down the middle from the top point, down in the middle of the triangle's base.