Isosceles triangle is acute, right or obtuse depends only on the angle at its apex. In Euclidean geometry, the base angles cannot be obtuse (greater than 90°) or right (equal to 90°) because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. Since a triangle is obtuse or right if and only if one of its angles is obtuse or right, respectively, an isosceles triangle is obtuse, right or acute if and only if its apex angle is respectively obtuse, right or acute. In Edwin Abbott's book "Flatland", this classification of shapes was used
Incircle and excircles of a triangle using the side lengths of the triangle relative to the perimeter—i.e., using the barycentric coordinates given above, normalized to sum to unity—as weights. (The weights are positive so the incenter lies inside the triangle as stated above.) If the three vertices are located at formula_14, formula_15, and formula_16, and the sides opposite these vertices have corresponding lengths formula_7, formula_8, and formula_9, then the incenter is at Denoting the incenter of triangle "ABC" as "I", the distances from the incenter to the vertices combined with the lengths of the triangle sides obey the equation Additionally, where "R" and "r" are the