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edge triangle

Computes the area of a trapezium within a triangle with a fixed edge.
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Edge Triangle

doubleh )[inline]
This module computes the area of the trapezium formed between a triangle with a fixed edge on a reference line and a line found at a given distance distance from this reference line.

This situation is described by the following image. The area which we want to compute is that of the filled trapezium \inline [BB_1C_1C].



Let \inline  \mathrm{xOy} be an orthogonal coordinate system and let \inline  \triangle ABC be an arbitrary triangle so that \inline  BC \subset \mathrm{Ox} and where \inline  a, b, c \in \mathbb{R}_+^* are fixed numbers. Also let \inline  d  \parallel \mathrm{Ox} so that the distance between the line \inline  d and \inline  \mathrm{Ox} equals \inline  h \in \mathbb{R}_+ and \inline  AB \cap d = \{B_1\}, \inline  AC \cap d = \{C_1\}.

Obviously \inline  \triangle AB_1C_1 \sim \triangle ABC which implies:

where H is the height corresponding to vertex A.

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Example 1

#include <codecogs/maths/geometry/area/edge_triangle.h>
#include <stdio.h>
int main()
  // the lengths of the sides
  double a = 3.3, b = 4.5, c = 5.4;
  // display the lengths of the sides
  printf("a = %.1lf\nb = %.1lf\nc = %.1lf\n\n", a, b, c);
  // display the area for different values of h
  for (double h = 0.1; h < 1.09; h += 0.1)
    printf("h = %.1lf   Area = %.2lf\n", h, 
    Geometry::Area::edge_triangle(a, b, c, h));
  return 0;

a = 3.3
b = 4.5
c = 5.4
h = 0.1   Area = 0.33
h = 0.2   Area = 0.65
h = 0.3   Area = 0.96
h = 0.4   Area = 1.26
h = 0.5   Area = 1.56
h = 0.6   Area = 1.85
h = 0.7   Area = 2.13
h = 0.8   Area = 2.40
h = 0.9   Area = 2.67
h = 1.0   Area = 2.93


The values of the sides must satisfy the triangle inequality.


afirst side of the triangle (BC)
bsecond side of the triangle (AC)
cthird side of the triangle (AB)
hthe distance between line \inline d and \inline \mathrm{Ox}


The value of the desired area.


Eduard Bentea (September 2006)
Source Code

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