1. Russian King Peter the Great announced that no
member of the Royal Family should marry until they pass in arithmetic, geometry
and navigation.
2. There is no equilateral triangle, the coordinates
of whose vertices are all integers.
3. There are only five Heronian triangles (triangles
having integer sides) such that the perimeter is equal to the area. They are:
(5, 12, 13) , (6, 8, 10) , (6, 25, 29) , (7, 15, 20) and (9, 10, 17). The
first two triangles are right triangles also.
4. A plane can be filled with squares, equilateral
triangles and regular hexagons. No other regular polygons can fill the plane
without gaps.
5. Three legged stool is more stable than any other
stool. The principle underlying is this fact is we can draw only one circle
through three given points which are not collinear. That is why old people walk
with the assistance of a stick (third leg) , or even think of tripod.
6. An angle of 3 degrees is the only constructible
angle of prime measure.
7. There are certainly at least two peoples in the
world with exactly the same number of hairs on their heads! (There are more
people than the hairs on any one head).
