Semiperimeter
The semiperimeter on a figure is defined as
 | (1) |
is the Perimeter. The semiperimeter of Polygons appears in unexpected ways in thecomputation of their Areas. The most notable cases are in the Altitude, Exradius, andInradius of a Triangle, the Soddy Circles, Heron's Formula for the Area of aTriangle in terms of the legs 
, 
, and 
 | (2) |
 | (3) |
For a Triangle, the following identities hold,
 |  |  | (4) |
 | |  | (5) |
 | |  | (6) |
Now consider the above figure. Let
be the Incenter of the Triangle 
, with 
, 
, and 
the tangent points of the Incircle.Extend the line 
with 
. Note that the pairs oftriangles 
, 
, 
are congruent. Then | |  | |
|  | ||
|  | ||
|  | ||
|  | (7) |
Furthermore,
| |  | |
|  | ||
|  | (8) | |
| |  | |
|  | ||
|  | (9) | |
| |  | (10) |
(Dunham 1990). These equations are some of the building blocks of Heron's
derivation of Heron's Formula.
References
Dunham, W. ``Heron's Formula for Triangular Area.'' Ch. 5 in Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 113-132, 1990.
- Lower Denjoy Sum
- Lower Integral
- Lower Limit
- Lower Sum
- Lower-Trimmed Subsequence
- Lowest Terms Fraction
- Loxodrome
- Lozenge
- Lozenge Method
- Lozi Map
- Lp'-Balance Theorem
- L-Polyomino
- L-Series
- L-System
- Lucas Correspondence
- Lucas Correspondence Theorem
- Lucas-Lehmer Residue
- Lucas-Lehmer Test
- Lucas' Married Couples Problem
- Lucas Number
- Lucas Polynomial
- Lucas Polynomial Sequence
- Lucas Pseudoprime
- Lucas Sequence
- Lucas's Theorem