The smallest integer-sided triangle with three rational medians is acute, with sides (68, 85, 87). There are no acute integer-sided triangles with area = perimeter, but there are three obtuse ones, having sides (6,25,29), (7,15,20), and (9,10,17). The only triangles with one angle being twice another and having integer sides in arithmetic progression are acute: namely, the (4,5,6) triangle and its multiples. The smallest-perimeter triangle with integer sides in arithmetic progression, and the smallest-perimeter integer-sided triangle with distinct sides, is obtuse: namely the one with sides (2, 3, 4). The only triangle with consecutive integers for an altitude and the sides is acute, having sides (13,14,15) and altitude from side 14 equal to 12. In any triangle, any two angle measures A and B opposite sides a and b respectively are related according to : p. The right triangle is the in-between case: both its circumcenter and its orthocenter lie on its boundary. Likewise, a triangle's circumcenter-the intersection of the three sides' perpendicular bisectors, which is the center of the circle that passes through all three vertices-falls inside an acute triangle but outside an obtuse triangle. These altitudesįall entirely outside the triangle, resulting in their intersection with each other (and hence with the extended altitude from the obtuse-angled vertex) occurring in the triangle's exterior.

But for an obtuse triangle, the altitudes from the two acute angles intersect only the extensions of the opposite sides. In the case of an acute triangle, all three of these segments lie entirely in the triangle's interior, and so they intersect in the interior. The orthocenter is the intersection point of the triangle's three altitudes, each of which perpendicularly connects a side to the opposite vertex.

However, while the orthocenter and the circumcenter are in an acute triangle's interior, they are exterior to an obtuse triangle.

In all triangles, the centroid-the intersection of the medians, each of which connects a vertex with the midpoint of the opposite side-and the incenter-the center of the circle that is internally tangent to all three sides-are in the interior of the triangle. 2.7 Distances involving triangle centers.2.6 Circumradius, inradius, and exradii.

Obtuse isosceles