MatheAss 9.0 − 2-dim. Geometry
Rectangular Triangles
If two sizes 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
From three outer quantities (sides or angles) of a triangle, the program calculates the sides, the angles, the heights, the side and angle halves, the circumference and the area content, as well as the centers and radii of the circle and perimeter 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 quantities (see triangles of three sizes).
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 center verticals, the side halves. the angle halves and the heights of a triangle. In addition, the centers and radii of the perimeter, the incircle and the three circles.
Given: ¯¯¯¯¯¯ Edges: A(1|0) B(5|1) C(3|6) Results: ¯¯¯¯¯¯¯ Vertices: 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.

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 content, the circumference and the coordinates of the corner and the surface center.
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 an 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