# MatheAss 9.0 − 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
```

## Triangles by three Elements

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 ri, circumcircle radius rc, 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:
¯¯¯¯¯¯¯
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
```