If a hexagon has vertices on the circumcircle of an acute triangle at the six points ( including three triangle vertices ) where the extended altitudes of the triangle meet the circumcircle, then the area of the hexagon is twice the area of the triangle.
12.
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.
13.
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.
14.
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.
15.
Every acute triangle has three inscribed squares ( squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle ).
16.
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.
17.
An acute triangle has three inscribed squares, each with one side coinciding with part of a side of the triangle and with the square's other two vertices on the remaining two sides of the triangle . ( In a right triangle two of these are merged into the same square, so there are only two distinct inscribed squares . ) However, an obtuse triangle has only one inscribed square, one of whose sides coincides with part of the longest side of the triangle.
18.
An acute triangle has three inscribed squares, each with one side coinciding with part of a side of the triangle and with the square's other two vertices on the remaining two sides of the triangle . ( A right triangle has only two distinct inscribed squares . ) If one of these squares has side length " x " " a " and another has side length " x " " b " with " x " " a " " b ", then
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