[Inscribed and Circumscribed Circles]
Is it necessarily true that two isosceles triangles are equal if their lateral sides and the radii of their inscribed circles are equal?
Solution
Consider "stretched horizontally" and "stretched vertically" triangles.
## Solution
Let's fix a circle of radius and consider isosceles triangles circumscribed around this circle. Such triangles are uniquely determined by the length of the height dropped to the base of the isosceles triangle. The length of this height can take any value in the interval from to infinity, and the length of the lateral side of the triangle continuously depends on the length of the height. If the height dropped to the base of the triangle is sufficiently large, then the lateral side is also sufficiently large. If the height is sufficiently small (i.e., it is "slightly more" than 2r), then the lateral side is also sufficiently large.
Therefore, if the value of the height is continuously changed from to infinity, then a fixed sufficiently
large value of the length of the lateral side will be achieved at two different values of the height.
## Answer
not necessarily.
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