MatheAss 10.0 − 2‑dim. Geometry

Right-angled triangles

If two of the following quantities 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

From three external quantities (sides or angles) of a triangle, the program calculates the sides, angles, heights, side and angle bisectors, the perimeter and the area, as well as the centers and radii of the incircle and circumcircle.

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 from three points

From the coordinates of three vertices, the program calculates all external and internal quantities (see Triangles from 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 lines and circles in a triangle (New in version 9.0)

The program determines the equations of the perpendicular bisectors, the side bisectors, the angle bisectors and the altitudes of a triangle. It also determines the centers and radii of the circumcircle, the incircle, the three excircles and the nine-point circle (from March 2025).

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 vertices and one of the following quantities are given, the program calculates the others.
Side a, incircle radius r_i, circumcircle radius r_u, 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 (since November 2022)

The program now also calculates the sides and angles of the polygon and checks whether the polygon is convex, concave or self‑intersecting.
In addition, convex polygons are checked for the existence of an incircle and/or a circumcircle.

Vertices:                       Area  A = 16
 A(1|2)                          
 B(4.5|0.5)                     Perimeter  u = 15.54498
 C(6|4)                           
 D(4.5|5.5)                     Vertex centroid: 
 E(1|4)                           VC(3.4|3.2)

                                     Area centroid: 
                                     AC(3.46875|3.07813)

Sides:                          Angles:
 |AB| = 3.8078866        ∠BAE = 113.19859°
 |BC| = 3.8078866        ∠CBA = 90°
 |CD| = 2.1213203        ∠DCB = 111.80141°
 |DE| = 3.8078866        ∠EDC = 111.80141°
 |EA| = 2                       ∠AED = 113.19859°

Cyclic polygon
Circumcircle:  M(3.5|3)  r = 2.6925824

Cyclic polygon:

Mappings (revised in version 9.0)

The program allows you to apply a chain of mappings to an n‑gon. You can choose from translation, reflection in a line, point reflection, rotation, central dilation and shear.

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

Translation: dx=2, dy=1  ☑
A₁(3|2), B₁(7|2), C₁(7|6), D₁(5|8), E₁(3|6)

Rotation: Z(2|-1), α = -60°  ☑
A₂(5.0981|-0.36603), B₂(7.0981|-3.8301),    
C₂(10.562|-1.8301), D₂(11.294|0.90192), 
E₂(8.5622|1.634)

Circle and circular sections

If two of the following quantities 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

Circle tangents (New in version 9.0 since February 2021)

The program calculates the equations of the following tangents:

The tangent to a circle k at 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 line g
The tangents to 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

Plane intersections

The program calculates the intersections of lines and circles

two lines

g : x + y = 0
h : x − y = 5

Intersection point : S(2.5|-2.5)

Angle of intersection: 90°

Distances from the origin:
  d(g,O) = 0
  d(h,O) = 3.5355339

line 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