1. In any triangle, the difference between any two sides is less than the third.
2. If any point within a triangle be joined to its angular points, the sum of the joining lines is greater than its semiperimeter.
3. If through the extremities of the base of a triangle, whose sides are unequal, lines be drawn to any point in the bisector of the vertical angle, their difference is less than the difference of the sides.
4. If the lines be drawn to any point in the bisector of the external vertical angle, their sum is greater than the sum of the sides.
5. Any side of any polygon is less than the sum of the remaining sides.
6. The perimeter of any triangle is greater than that of any inscribed triangle, and less than that of any circumscribed triangle.
7. The perimeter of any polygon is greater than that of any inscribed, and less than that of any circumscribed, polygon of the same number of sides.
8. The perimeter of a quadrilateral is greater than the sum of its diagonals.
Def.—A line drawn from any angle of a triangle to the middle point of the opposite side is called a median of the triangle.
9. The sum of the three medians of a triangle is less than its perimeter.