No matter how you choose to place the red points on the circumference, the three blue points will lie on a straight line (shown in red). This works for any conic (which may be a circle, ellipse, parabola or hyperbola).
It is conjectured that n is a sum of 3 cubes if n is a number that is not congruent to 4 or 5 mod 9. The number 33 enters this category, but for 64 years no solutions emerged — that is, whether the equation 33 = x³ + y³ + z³ has an integer solution. Continue reading ““Stubborn” Number 33″
Given 3 circles, each intersecting the other two in two points, the line segments connecting their points of intersection satisfy: ace/bdf = 1
In any triangle, the 3 points of intersection of the adjacent angle trisectors ALWAYS form an equilateral triangle (in blue), called the Morley triangle.