structural:scholar_edition, stylistic:tone
Characterizations

A triangle △ A B C {\displaystyle \triangle ABC} with sides a ≤ b < c {\displaystyle a\leq b<c}, semiperimeter s = 1 2 (a + b + c) {\textstyle s={\tfrac {1}{2}}(a+b+c)}, area T, {\displaystyle T,} altitude h c {\displaystyle h_{c}} opposite the longest side, circumradius R, {\displaystyle R,} inradius r, {\displaystyle r,} exradii r a, r b, r c {\displaystyle r_{a},r_{b},r_{c}} tangent to a, b, c {\displaystyle a,b,c} respectively, and medians m a, m b, m c {\displaystyle m_{a},m_{b},m_{c}} is a right triangle if and only if any one of the statements in the following six categories is true. Each of them is thus also a property of any right triangle.