Can we define the apothem for any triangle?

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Is there an apothem for any triangle, because every triangle can be circumscribed in the circumference, and so we have the radius of the circumference inscribed? Is there or is there not an apothem for a scalene triangle, for example?

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2 Answers

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The term apothem is usually (always?) reserved for regular polygons, so among triangles the term only applies to equilateral ones.

The radius of the circumscribing circle of any triangle (or more generally, any cyclic polygon) is called the circumradius, but this never coincides with the apothem. Rather, for equilateral triangles the apothem coincides with the inradius, which is defined for any triangle (or other applicable polygon) as the radius of the unique inscribed circle.

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Regular polygons are the only polygons which have apothems; therefore there will be no apothem in a scalene triangle. Thus, there is not an apothem in any triangle. $\blacktriangle$

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