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A magic triangle or perimeter magic triangle[1] is an arrangement of the integers from 1 to n on the sides of a triangle with the same number of integers on each side, called the order of the triangle, so that the sum of integers on each side is a constant, the magic sum of the triangle. [1][2][3][4] Unlike magic squares, there are different ...
Magic circles were invented by the Song dynasty (960–1279) Chinese mathematician Yang Hui (c. 1238–1298). It is the arrangement of natural numbers on circles where the sum of the numbers on each circle and the sum of numbers on diameters are identical. One of his magic circles was constructed from the natural numbers from 1 to 33 arranged ...
The apparent triangles formed from the figures are 13 units wide and 5 units tall, so it appears that the area should be S = 13×5 / 2 = 32.5 units. However, the blue triangle has a ratio of 5:2 (=2.5), while the red triangle has the ratio 8:3 (≈2.667), so the apparent combined hypotenuse in each figure is actually bent.
Miquel's theorem. Miquel's theorem is a result in geometry, named after Auguste Miquel, [1] concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides. It is one of several results concerning circles in Euclidean geometry due to Miquel, whose work was published in Liouville's ...
The primary square is obtained by rotating the root square counter-clockwise by 90 degrees, and replacing the numbers. The resulting square is an associative magic square, in which every pair of numbers symmetrically opposite to the center sum up to the same value, 26. For e.g., 16+10, 3+23, 6+20, etc.
The largest circle (curvature k 4) may also be replaced by a smaller circle with positive curvature ( k 0 = 4pp' − qq' ). EXAMPLE: Using the area and four radii obtained above for primitive triple [44, 117, 125] we obtain the following integer solutions to Descartes' Equation: k 1 = 143 , k 2 = 99 , k 3 = 26 , k 4 = (−18) , and k 0 = 554 .
In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826. [1] Specifically, the power of a point with respect to a circle with center and radius is defined by. If is outside the circle, then , if is on the circle, then ...
Li Ye's inscribed circle in triangle:Diagram of a round town Yang Hui's magic concentric circles – numbers on each circle and diameter (ignoring the middle 9) sum to 138 Ceyuan haijing ( Chinese : 測圓海鏡 ; pinyin : Cèyuán Hǎijìng ), or Sea-Mirror of the Circle Measurements , is a collection of 692 formula and 170 problems related to ...