A polygon semi - perimeter is half its perimeter . Although the semi-perimeter is a very simple derivative of the perimeter, it appears so often in formulas for triangles and other geometric figures that it was given a separate name. If the semi-perimeter is in any formula, it is usually denoted by the letter p .
Content
Triangles
The semi-perimeter is most often used for triangles. The formula of a semi-perimeter for a triangle with sides a , b and c
Properties
In any triangle, the vertex and the point of tangency of the extra-written circle on the opposite side divide the perimeter of the triangle into two equal parts, that is, into two paths, the length of each of which is equal to the half-perimeter. The figure shows the sides A, B, C and the tangency points A ', B', C ' , then
Three segments connecting vertices with opposite tangency points intersect at one point - the Nagel point .
If we consider the segments connecting the midpoints of the sides with the points separated (along the sides) from this midpoint by a semi-perimeter, then these segments intersect at one point - the center of the Speaker circle , which is a circle inscribed in the . The center of Spear is the center of gravity of the sides of the triangle.
The straight line passing through the center of the inscribed circle of the triangle divides the perimeter in half if and only if it divides the area in half.
The semi-perimeter of a triangle is equal to the perimeter of its .
It follows from the triangle inequality that the length of the longest side of the triangle does not exceed a semi-perimeter.
Formulas with semi-perimeter
The area K of any triangle is the product of the radius of its inscribed circle and semi-perimeter:
The area of a triangle can be calculated from its semi-perimeter and the side lengths a, b, c using the Heron formula :
The radius of the circumscribed circle R of a triangle can also be calculated from its semi-perimeter and side lengths:
This formula can be derived from the sine theorem .
The radius of the inscribed circle is
The cotangents theorem gives the cotangents of half angles at the vertices of a triangle in terms of a semi-perimeter, sides and radius of the inscribed surroundings.
The length of the bisector of the inner angle opposite to side a is [1]
In a right-angled triangle, the radius of an extra-inscribed circle on the hypotenuse is equal to a semi-perimeter. The semi-perimeter is equal to the sum of the inscribed circle radius and the double radius described. The area of a right triangle is where a and b are the legs.
Quadrilaterals
Formula for a semi-perimeter quadrilateral with sides a , b , c and d
One of the formulas for triangles using a semi-perimeter is also applicable to the described quadrilaterals , which have an inscribed circle and the sum of the lengths of the opposite sides of which is equal to the semi-perimeter. Namely, this is the formula area of the figure:
The simplest form of the Brahmagupta formula for the area of a quadrilateral inscribed in a circle has the form close to the Heron formula for the area of a triangle:
The Bretschneider relation generalizes the formula for all convex quadrilaterals:
Where and - two opposite corners.
The four sides of the are four solutions of an equation of the fourth degree, the parameters of which are a semi-perimeter, the radius of the inscribed circle, and the radius described.
Regular polygons
The area of a convex regular polygon is equal to the product of its semi-perimeter and the distance from the center to one of the sides.
Notes
- ↑ Johnson, 2007 , p. 70
Literature
- Roger A. Johnson. Advanced Euclidean Geometry. - Dover Publ., 2007. (Reissue of the 1929 book)
Links
- Weisstein, Eric W. Semiperimeter (English) on the Wolfram MathWorld website.