> MatheAss − Special lines in a triangle

Special lines in a triangle

If the coordinates of the three vertices of a triangle are entered, the program calculates the equations of the perpendicular bisectors[1], of the side bisectors[2], of the angle bisectors[3] and of the hauteurs[4]. In addition, the centers and radii of the circumcircle[5], of the incircle[6], of the three excircles[7] and the nine-point circle [8].

A list of check boxes can be used to select which objects are to be calculated and drawn.

Perpendicular bisec.
Medians
Angle bisectors
hauteurs
Incircle
Circumcircle
Excircles
Nine-point circle

Example 1: incircle and excircles of a triangle

Given:
¯¯¯¯¯¯
   Vertices:    A(1|0)   B(5|1)   C(3|6)
 
Results:
¯¯¯¯¯¯¯
       Sides:   a :  5·x + 2·y = 27
                    b :  3·x - y = 3
                    c :  x - 4·y = 1
 
   Incircle:    Mi(3,119|1,962)         r i = 1,390  
 
   Excircles: Ma(7,626|6,136)       ra = 4,346
                   Mb(-4,356|5,784)      rb = 6,910
                   Mc(3,248|-2,427)      rc = 2,900

The center of the incircle (green) lies on the bisector of the three interior angles. The centers of the excircles (red) are each on the bisector of an inner angle and on the bisector of the outside angle of the other two triangle angles. These construction lines are also drawn.

Example 2: hauteurs in an obtuse-angled triangle

Given:
¯¯¯¯¯¯
    Vertices:  A(7|3)   B(16|10)  C(8|9)
 
Results:
¯¯¯¯¯¯¯
          Sides:   a : -x + 8·y = 64
                        b : 6·x - y = 39
                        c : 7·x - 9·y = 22
 
      hauteurs:  ha : 8·x + y = 59
                       hb : x + 6·y = 76
                       hc : 9·x + 7·y = 135
  
hauteurs feet: Ha(6,277|8,785)  
                       Hb(8,378|11,27)
                       Hc(10,53|5,746)

 Orthocenter: H(11,05|8,26)

The intersection of the hauteurs of an obtuse-angled triangle lies outside the triangle. The construction lines are also drawn. In order to make them more visible, the grid lines have been hidden.

Example 3: Excircles and nine-point circle

Given:
¯¯¯¯¯¯
          Vertices:    A(1|0)   B(5|1)   C(3|6)
 
Results
¯¯¯¯¯¯¯
              Sides:   a :  5·x + 2·y = 27
                           b :  3·x - y = 3
                           c :  x − 4·y = 1
 
          Excircles:  Ma(7,626|6,136)       ra = 4,346
                            Mb(-4,356|5,784)     rb = 6,91
                            Mc(3,248|-2,427)     rc = 2,9

 Nine-point circle: M9(3,295|2,068)      r9 = 1,596

The nine-point circle touches the incircle and the excircles (Feuerbach's theorem)..

See also:

Setting the graphics
Wikipedia: Incircle and excircles | Nine point circle

[1]  The perpendicular bisectors are the straight lines, which intersect one side of the triangle vertically at its center. Their intersection point is the center of the perimeter.
[2]  The side bisectors are the line segments from an edge of the triangle to the center of the opposite side. Their intersection point is the center of gravity of the triangle.
[3]  The angle bisectors, as their name suggests, halves one of the inner angles of the triangle. Their intersection point is the center of the triangle's incircle.
[4]  The hauteurs are the plumb lines from one corner of the triangle to the opposite side. Their intersection point is called the triangle's orthocenter.
[5]  The circumcircle of a Triangle is the circle through the edges of the triangle. In an acute-angled triangle, its center lies inside, in an obtuse-angled triangle it lies outside the triangle.
[6]  The incircle is the circle that touches the three sides of the triangle from the inside. Its center is the intersection of the angle bisectors.
[7]  The excircles touch one triangle side from the outside and are tangentially touched by the extensions of the other two sides.
[8]  The nine-point circle is a circle that passes through nine significant points of the triangle. These are the midpoint of each side, the foot of each altitude and the midpoint of the upper line segment of each altitude.