Likewise, a triangle's circumcenter & mdash; the intersection of the three sides'perpendicular bisectors, which is the center of the circle that passes through all three vertices & mdash; falls inside an acute triangle but outside an obtuse triangle.
12.
Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called " acute " if its apex is above the interior of the base and " obtuse " if its apex is above the exterior of the base.
13.
Even if a polygon has a circumscribed circle, it may not coincide with its minimum bounding circle; for example, for an obtuse triangle, the minimum bounding circle has the longest side as diameter and does not pass through the opposite vertex.
14.
For an acute triangle, six of the points ( the midpoints and altitude feet ) lie on the triangle itself; for an obtuse triangle two of the altitudes have feet outside the triangle, but these feet still belong to the nine-point circle.
15.
In an obtuse triangle ( one with an obtuse angle ), the foot of the altitude to the obtuse-angled vertex falls on the opposite side, but the feet of the altitudes to the acute-angled vertices fall on the opposite extended side, exterior to the triangle.
16.
The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is 2 \ sqrt { 2 } / 3 = 0.94 . . . . Both of these extreme cases occur for the isosceles right triangle.
17.
As a result, in an isosceles triangle with one or two angles of 36? the longer of the two side lengths is " ? " times that of the shorter of the two, both in the case of the acute as in the case of the obtuse triangle.
18.
In these cases the vertex figure is highly distorted to achieve uniformity with flat faces : in the first two cases it is an obtuse triangle instead of an acute triangle, and in the last three it is a pentagram or hexagram instead of a pentagon or hexagon, winding around the centre twice.
19.
Since barycentric coordinates are all positive for a point in a triangle's interior but at least one is negative for a point in the exterior, and two of the barycentric coordinates are zero for a vertex point, the barycentric coordinates given for the orthocenter show that the orthocenter is in an acute triangle's interior, on the right-angled vertex of a right triangle, and exterior to an obtuse triangle.
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