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In any cyclic quadrilateral, any two opposite sides are antiparallel with respect to the other two sides. Lines l 1 {\displaystyle l_{1}} and l 2 {\displaystyle l_{2}} are antiparallel with respect to the line m {\displaystyle m} if they make the same angle with m {\displaystyle m} in the opposite senses.
Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in general parallel. Instead, each pair of sides is antiparallel with respect to the other, with sides in the longer pair crossing each other as in a scissors mechanism. Whereas a parallelogram's opposite angles are equal ...
In today's puzzle, there are eight theme words to find (including the spangram). Hint: The first one can be found in the bottom half of the board. Here are the first two letters for each word: SK ...
For every concave kite there exist two circles tangent to two of the sides and the extensions of the other two: one is interior to the kite and touches the two sides opposite from the concave angle, while the other circle is exterior to the kite and touches the kite on the two edges incident to the concave angle. [28]
Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite. [5] However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms ...
One side (left in the illustration) is partially folded in half and pinched to leave a mark. The intersection of a line from this mark to an opposite corner (red) with a diagonal (blue) is exactly one third from the bottom edge. The top edge can then be folded down to meet the intersection. [4]
A somewhat easier (more symmetrical) puzzle, the 8×8 rectangle with a 2×2 hole in the center, was solved by Dana Scott as far back as 1958. [10] There are 65 solutions. Scott's algorithm was one of the first applications of a backtracking computer program. Variations of this puzzle allow the four holes to be placed in any position.
In a triangle, any arbitrary side can be considered the base. The two endpoints of the base are called base vertices and the corresponding angles are called base angles. The third vertex opposite the base is called the apex. The extended base of a triangle (a particular case of an extended side) is the line that contains the base.
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