It is the shape formed by three non-collinear points on the surface of a sphere which are joined together by straight lines in which two of the sides are of the same length.
An example might be a triangle from the earth's North pole to two points on the same latitude - specially the equator.
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On the surface of a sphere, find any three points which are not on a great circle. The shape formed by joining the three pairs of points using arcs of great circles is a spherical triangle. The triangle is isosceles if two of the arcs are of the same length [and the third is not].
A great circle, in this context, is a circle on the surface of the earth and whose centre is at the centre of the earth.
Yes, they do exist.
A triangle that has one right angle and is isosceles is called an isosceles right triangle.
Isosceles
An isosceles triangle can have a 90 degree angle, but doesn't necessarily. Similarly, a right triangle can be an isosceles triangle, but isn't necessarily.
There can be only one pair of perpendicular lines in an isosceles triangle when it is a right angled isosceles triangle. A non right angled isosceles triangle has no pairs of perpendicular lines.