The orthocenter is the intersection point of the triangle's three altitudes, each of which perpendicularly connects a side to the opposite vertex.
But for an obtuse triangle, the altitudes from the two acute angles intersect only the extensions of the opposite sides.
The right triangle is the in-between case: both its circumcenter and its orthocenter lie on its boundary.
In any triangle, any two angle measures A and B opposite sides a and b respectively are related according to[1]: p. 264 This implies that the longest side in an obtuse triangle is the one opposite the obtuse-angled vertex.
115 All triangles in which the Euler line is parallel to one side are acute.
Since an acute angle has a positive tangent value while an obtuse angle has a negative one, the expression for the product of the tangents shows that for acute triangles, while the opposite direction of inequality holds for obtuse triangles.
Also, an acute triangle satisfies[4]: p.26, #954 in terms of the excircle radii ra , rb , and rc , again with the reverse inequality holding for an obtuse triangle.
If one of the inscribed squares of an acute triangle has side length xa and another has side length xb with xa < xb, then[2]: p. 115 If two obtuse triangles have sides (a, b, c) and (p, q, r) with c and r being the respective longest sides, then[4]: p.29, #1030 The Calabi triangle, which is the only non-equilateral triangle for which the largest square that fits in the interior can be positioned in any of three different ways, is obtuse and isosceles with base angles 39.1320261...° and third angle 101.7359477...°.
[5] The heptagonal triangle, with sides coinciding with a side, the shorter diagonal, and the longer diagonal of a regular heptagon, is obtuse, with angles
[6] There are no acute integer-sided triangles with area = perimeter, but there are three obtuse ones, having sides[7] (6,25,29), (7,15,20), and (9,10,17).
The smallest integer-sided triangle with three rational medians is acute, with sides[8] (68, 85, 87).
The oblique Heron triangle with the smallest perimeter is acute, with sides (6, 5, 5).