How can an isosceles triangle be identified?
Isosceles triangles can be identified by its two independent elements, like a side and an angle at the base or a base and an altitude etc.
While a general triangle requires three elements to be fully identified, an isosceles triangle requires only two because we have the equality of its two sides and two angles.
Notice that if you can construct a unique triangle using given elements, these elements fully define a triangle.
As an example, you can consider any construction problem that involves two given elements, like
(a) construct an isosceles triangle by a base and an opposite angle;
(b) same by a median to a base and an angle between a base and a side;
(c) same by a radius of a circumscribing circle and an angle at the top;
Here is a solution to one such construction problem.
Construct an isosceles triangle by a radius of an inscribed circle and a base.
1. Draw a circle using a given radius.
2. Draw a tangent to this circle.
3. From a common point between a circle and a tangent along the tangent mark two points on a distance equaled to one half of the given base.
4. From two points obtained at step 3 draw two tangent to the same circle.
The above solution is possible only if the base is greater than a diameter of an inscribed circle, otherwise the problem has no solution.
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