MatheAss 10.0 − 2-dim. Geometry

Rectangular Triangles

If two properties of a rectangular triangle are given, the program calculates the others.

Given:
¯¯¯¯¯¯
Hypot. segment  p = 1,8
                  Area  A = 6 
 
Results :
¯¯¯¯¯¯¯
              Cathete a = 3
              Cathete b = 4
       Hypotenuse c = 5
                Angle  α = 36,869898°
                Angle  β = 53,130102°  
Hypot. segment  q = 3,2
             Altitude  h = 2,4

Given three outer properties (sides or angles) of a triangle, the program calculates the sides, the angles, the altitudes, the medians and the angle bisectors, the perimeter and the area, as well as the centers and the radii of the incircle and the circumcircle of the triangle.

Given:  a=6, b=4 and α=60°

   Vertices : A(1|1)         B(7,899|1)     C(3|4,4641)    
      Sides : 6                  4                    6,89898        
    Angles : 60°               35,2644°       84,7356°       
 Altitudes : 3,98313       5,97469         3,4641         
 Medians : 4,77472       6,148             3,75513        
   Bisectr. : 4,38551       6,11664         3,5464         

Circumcir.: M(4,44949|1,31784)       ru = 3,4641
    Incircle : O(3,44949|2,41421)       r i = 1,41421

       Area : A = 11,9494   Perimeter : u = 16,899

Triangles of three Points

From the coordinates of three vertices, the program calculates all outer and inner properties (see Triangles by three elements ).

  Vertices :  A(1|0)          B(5|1)           C(3|6)         
     Sides :  5,38516        6,32456       4,12311        
    Angles :  57,5288°      82,2348°     40,2364°       
 Altitudes :  4,0853         3,47851        5,33578        
 Medians :  4,60977       3,60555        5,5            
   Bisectr. :  4,37592       3,51849        5,46225        

Circumcir.:  M(2,40909|2,86364)       ru = 3,19154
    Incircle :  O(3,11866|1,96195)       r i = 1,38952

         Area :  A = 11           Perimeter : u = 15,8328

Special Straight Lines in a Triangle (New in version 9.0)

The program determines the equations of the perpendicular bisectors, of the medians, of the angle bisectors and of the altitudes of a triangle. In addition, the centers and radii of the circumcircle, the incircle and the three excircles.

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

Regular Polygons

If the number of corners and one of the following sizes are given, the program calculates the others.
Side a, incircle radius r_i, circumcircle radius r_c, perimeter u or area A.

Given:
¯¯¯¯¯¯ 
       Vertices  n = 6
Circumcircle rc = 1
 
Results:
¯¯¯¯¯¯¯ 
           Side  a = 1
        Incircle ri = 0,8660254
   Perimeter  p = 6
           Area  A = 2,5980762

Arbitrary Polygons

From the coordinates of the vertices of a polygon, the program calculates the area, the perimeter, the centroid of vertices and the centroid of area.

Vertices:          Area  A = 18
 A(0|0)     
 B(4|1)     Perimeter  p = 22,032567
 C(6|0)     
 D(5|7)     Centroid of vertices:
                CV(3,75|2)
            
                Centroid of area:
                CA(3,72222|2,66667)

Mappings of Polygons (revised in version 9.0)

The program makes it possible to apply a concatenation of mappings to a polygon. You can choose from displacement, straight reflection, point reflection, rotation, centric stretching and shear.

Original polygon
A(1|1), B(5|1), C(5|5), D(3|7), E(1|5), 

1. Translation: dx=2, dy=1  ☑
A(3|2), B(7|2), C(7|6), D(5|8), E(3|6), 

2. Rotation: Z(2|-1), α=-60° ☑
A(5,0981|-0,36603), B(7,0981|-3,8301),
C(10,562|-1,8301), D(11,294|0,90192),
E(8,5622|1,634),

Circular Sections

If two of the following sizes are given, the program calculates the others.

Given:
¯¯¯¯¯¯ 
                 Arc b = 1
             Angle α = 45°

Results:  
¯¯¯¯¯¯¯ 
           Radius r = 1,2732395
            Chord s = 0,97449536
       Section A1 = 0,63661977
       Distance d = 1,17632
 Arrow height h = 0,096919589
    Segment A2 = 0,063460604

             Area A = 5,0929582
     Perimeter p = 8

Tangent Lines to Circles (New in version 9.0 from February 2021)

The equations of the following tangents are calculated:

The tangent to a circle k in a point B
The tangents to a circle k through a point P outside the circle
The tangents to a circle k parallel to a straight line g
The tangents on two circles k₁ and k₂

Given:
¯¯¯¯¯
  k1 : M(5|8) ,    r=5
  k2 : M(-1|2) ,   r=3

Outer tangents
¯¯¯¯¯¯¯¯¯¯¯¯
  t₁: -4,2923·x + 7,04104·y = -6,36427
  t₂: -7,04104·x + 4,29230·y = 40,3643

Inner tangents
¯¯¯¯¯¯¯¯¯¯¯¯
  t₃: 1,21895·x + 2,55228·y = 12,3709
  t₄: -2,55228·x - 1,21895·y = -8,3709

Intersections in the Plane

The program calculates the intersections of straight lines and circles

Two Straights

g : x + y = 0
h : x - y = 5

Intersection point : S(2,5|-2,5)

Intersection angle : 90°

Distances from origin :
  d(g,O) = 0
  d(h,O) = 3,5355339

Straight and Circle

Circle and line :
¯¯¯¯¯¯¯¯¯¯¯¯¯
  k : M(5|0)   r = 5
  g : x + y = 0

Intersection points :      
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  S1(5|-5)
  S2(0|0)

Two Circles

Given are the circles :         
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 
k1 : M1(5|5) r1 = 5
k2 : M2(0|0) r2 = 5

Intersection points :
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 
S1(5|0)   S2(0|5)

Connecting line :
¯¯¯¯¯¯¯¯¯¯¯¯¯
 x + y = 5

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