Equivalently, two radii are parallel to two sides of the parent triangle, as will be shown in examples 1 and 2 below.
Everything we’ve been considering so far seems to work smoothly for the -triangle.
Since the parent triangle is obtuse, its circumcenter lies outside, opposite the obtuse angle. Thus is a (convex) quadrilateral.
If is the radius of the circumcircle of triangle , then we have that and (see here). Since is the circumcenter, we also have that . Hence, is a rhombus; in fact, a special rhombus with the property that the diagonal is equal to the equal sides. Every rhombus is a parallelogram so we conclude that radius is parallel to side and radius is parallel to side .
Suppose that a triangle with circumcenter has the property that radius is parallel to side and radius is parallel to side . Such a triangle is necessarily obtuse, so that the circumcenter lies outside the triangle and we again have a quadrilateral . Join and let . Since is parallel to , we have that . Since is the circumcenter, triangle is isosceles and so . The situation is shown below, with the circumcircle included:
Since is parallel to , we have that . Thus, triangle is isosceles; moreover, . By the inscribed angle theorem:
Thus, the interior angles of triangle are and .
Since the area satisfies , we have that .
Normally, . And so we must have . In turn, this implies
is not possible due to the fact that and . So we take , which yields . Using we get
No matter how the vertices are arranged, we’ll end up with a rhombus in which a diagonal is equal to the side-lengths. The interior angles of such a rhombus are . Now just consider the triangle that remains when is removed.
Consider various arrangements of the rhombus , for example the one shown below:
Apply the inscribed angle theorem to see that it is not possible to have be a square.
- (Aufbau) In triangle , let be the side-lengths, the circumradius, the circumcenter, the nine-point center, and the orthocenter. PROVE that the following statements are equivalent:
- the reflection of over is
- the area satisfies
- the circle with diameter passes through vertices and
- radius is parallel to side and radius is parallel to side
- the reflection of over is and the reflection of over is
- is the geometric mean of and , and is the geometric mean of and . ( and are the feet of the altitudes from and , respectively.)