holding for equilateral triangles. If we denote that the orthocenter divides one altitude into segments of lengths u and v, another altitude into segment lengths w and x, and the third altitude into segment lengths y and z, then uv = wx = yz. The distance from a side to the circumcenter equals half the distance from the opposite vertex to the orthocenter. The sum of the squares of the distances from the vertices to the orthocenter H plus the sum of the squares of the sides equals twelve times the square of the circumradius: Angles In addition to the law of sines, the law of cosines, the law of tangents, and the trigonometric existence conditions given earlier, for any triangle Morley's trisector theorem Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the Morley triangle. Figures inscribed in a triangle Conics As discussed above, every triangle has a unique inscribed circle (incircle) that is interior to the triangle and tangent to all three sides. Every triangle has a unique Steiner inellipse which is interior to the triangle and tangent at the midpoints of the sides. Marden's theorem shows how to find the foci of this ellipse. This ellipse has the greatest area of any ellipse tangent to all three sides of the triangle. The Mandart inellipse of a triangle is the ellipse inscribed within the triangle tangent to its sides at the contact points of its excircles. For any ellipse inscribed in a triangle ABC, let the foci be P and Q. Then Convex polygon Every convex polygon with area T can be inscribed in a triangle of area at most equal to 2T. Equality holds (exclusively) for a parallelogram. Hexagon The Lemoine hexagon is a cyclic hexagon with vertices given by the six intersections of the sides of a triangle with the three lines that are parallel to the sides and that pass through its symmedian point. In either its simple form or its self-intersecting form, the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle. Squares 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