**The geometry of the triangle**

**Romanova Elena, 8 class, school 57, Kazan, Russia**

Construction of triangle and calculation its angles

Construction of bisectors

Construction of medians

Construction of altitudes

> restart:with(geometry):

**The setting of the height of the triandle and let's call it "Т"**

> triangle(T,[point(A,4,6),point(B,-3,-5),point(C,-4,8)]);

T

** Construction of the triangle**

> draw(T,axes=normal,view=[-8..8,-8..8]);

**Construction of the triangle АВС**

> draw({T(color=gold,thickness=3)},printtext=true,axes=NONE);

**Calculation of the distance between heights А and В - the length of a side АВ**

> d1:=distance(A,B);

d1 := sqrt(170)

**Calculation of the distance between heights В and С - the length of a side ВС**

> d2:=distance(B,C);

d2 := sqrt(170)

**The setting of line which passes through two points А and В**

> line(l1,[A,B]);

l1

**Display the equation of line l1**

> Equation(l1);

> x;

> y;

-2 + 11 x - 7 y = 0

**The setting of line which passes through two points А and С**

> line(l2,[A,C]);

l2

** Display the equation of line l2**

> Equation(l2);

> x;

> y;

56 - 2 x - 8 y = 0

** The setting of line which passes through two points В and С**

> line(l3,[B,C]);

l3

**Display the equation of line l3**

> Equation(l3);

> x;

> y;

-44 - 13 x - y = 0

** Check the point А lies on line l1**

> IsOnLine(A,l1);

true

**Check the point А lies on line l1**

> IsOnLine(B,l1);

true

** Calculation of the andle between lines l1 and l2**

> FindAngle(l1,l2);

arctan(3)

** The conversion of result to degrees**

> b1:=convert(arctan(97/14),degrees);

97

arctan(--) degrees

14

b1 := 180 ------------------

Pi

**Calculation of decimal value of this angle**

> b2:=evalf(b1);

b2 := 81.78721981 degrees

**Calculation of the andle between lines l1 and l3**

> FindAngle(l1,l3);

arctan(3/4)

**The conversion of result to degrees**

> b3:=convert(arctan(97/99),degrees);

97

arctan(--) degrees

99

b3 := 180 ------------------

Pi

**Calculation of decimal value of this angle**

> b4:=evalf(b3);

b4 := 44.41536947 degrees

**Calculation of the angle between lines l2 and l3**

> FindAngle(l2,l3);

arctan(3)

**The conversion of result to degrees**

> b5:=convert(arctan(97/71),degrees);

97

arctan(--) degrees

71

b5 := 180 ------------------

Pi

**Calculation of decimal value of this angle**

> b6:=evalf(b5);

b6 := 53.79741070 degrees

**Check the sum of all the angles of the triangle**

> b2+b4+b6;

180.0000000 degrees

** Analytical information about the point А**

> detail(A);

name of the object: A

form of the object: point2d

coordinates of the point: [4, 6]

** Analytical information about the point В**

> detail(B);

name of the object: B

form of the object: point2d

coordinates of the point: [-3, -5]

**Analytical information about the point С**

> detail(C);

name of the object: C

form of the object: point2d

coordinates of the point: [-4, 8]

**The setting of heights of the triangle points A,B,C and let's call it "Т"**

with(geometry):

> triangle(ABC, [point(A,7,8), point(B,6,-7), point(C,-6,7)]):

**The setting of the bisector of angle А in triandle АВС**

> bisector(bA, A, ABC);

bA

**Analytical information about the bisector of angle А in the triandle**

> detail(bA);

name of the object: bA

form of the object: line2d

assume that the name of the horizonal and vertical axis are _x and _y

equation of the line: (15*170^(1/2)+226^(1/2))*_x+(-13*226^(1/2)-170^(1/2))*_y+97*226^(1/2)-97*170^(1/2) = 0

** Construction of the triangle**

> draw(ABC,axes=normal,view=[-8..8,-8..8]);

**Construction of the triangle ABC**

> draw({ABC(color=gold,thickness=3)},printtext=true,axes=NONE);

**Construction of the bisector of angle А**

> draw({ABC(color=gold,thickness=3),bA(color=green,thickness=3)},printtext=true,axes=NONE);

**The setting of the bisector of angle В in the triangle АВС**

> bisector(bB, B, ABC);

bB

**Analytical information about the bisector of angle B in the triandle**

> detail(bB);

name of the object: bB

form of the object: line2d

assume that the name of the horizonal and vertical axis are _x and _y

equation of the line: (-15*340^(1/2)-14*226^(1/2))*_x+(-12*226^(1/2)+340^(1/2))*_y+97*340^(1/2) = 0

**Construction of the bisector of angle В**

>draw({ABC(color=gold,thickness=3),bA(color=green,thickness=3),bB(color=red,thickness=3)},printtext=true,axes=NONE);

** The setting of the bisector of angle С in the triangle АВС**

> bisector(bC, C, ABC);

bC

**Analytical information about the bisector of angle С in the triangle**

> detail(bC);

name of the object: bC

form of the object: line2d

assume that the name of the horizonal and vertical axis are _x and _y

equation of the line: (14*170^(1/2)-340^(1/2))*_x+(13*340^(1/2)+12*170^(1/2))*_y-97*340^(1/2) = 0

**Construction of the bisector of angle С**

>draw({ABC(color=gold,thickness=3),bA(color=green,thickness=3),bB(color=red,thickness=3),bC(color=blue,thickness=3)},printtext=true,axes=NONE);

** Calculation of the point of intersection of the bisectors and let's call it "О"**

> intersection(O,bA,bB,bC);coordinates(O);

O

7 sqrt(85) - 3 sqrt(2) sqrt(113) + 3 sqrt(85) sqrt(2)

[2 -----------------------------------------------------,

sqrt(85) sqrt(2) + sqrt(2) sqrt(113) + 2 sqrt(85)

-16 sqrt(85) - 7 sqrt(2) sqrt(113) + 7 sqrt(85) sqrt(2)

- -------------------------------------------------------]

sqrt(85) sqrt(2) + sqrt(2) sqrt(113) + 2 sqrt(85)

**Construction of the bisectors and marking of the point of intersection "О" in the triandle**

>draw({ABC(color=gold,thickness=3),bA(color=green,thickness=3),bB(color=red,thickness=3),bC(color=blue,thickness=3),O},printtext=true,axes=NONE);

> restart:

> with(geometry):

**The setting of the heights of the triangle points A,B,C and let's call it "Т"**

> point(A,7,8),point(B,6,-7),point(C,-6,7);

A, B, C

** Let's call "Т1"**

> triangle(T1,[A,B,C]);

T1

** Construction of "Т1"**

> draw(T1(color=gold,thickness=3),axes=NONE,printtext=true);

**The setting of the median from the point В in the trianglemedian(mB,B,T1,B1);**

> median(mb,B,T1);

mB

mb

** Construction of the median from the point В**

> draw({T1(color=gold,thickness=3),mB(color=green,thickness=3),mb},printtext=true,axes=NONE);

**The setting of the median from the point А in the trianglemedian(mA,A,T1,A1);**

> median(ma,A,T1);

mA

ma

** Construction of the median from the point А**

>draw({T1(color=gold,thickness=3),mB(color=green,thickness=3),mA(color=magenta,thickness=3),ma},printtext=true,axes=NONE);

**The setting of the median from the point С in the trianglemedian(mC,C,T1,C1);**

> median(mc,C,T1);

mC

mc

**Costruction of the median from the point С**

>draw({T1(color=gold,thickness=3),mB(color=green,thickness=3),mA(color=magenta,thickness=3),mA,mC(color=maroon,thickness=3)},printtext=true,axes=NONE);

**Calculation of the point of intersection of the median and let's call it "О"**

>intersection(O,ma,mb,mC);coordinates(O);

O

[7/3, 8/3]

** Construction of medians and marking of the point of intersection "О" in the triangle**

>draw({T1(color=gold,thickness=3),mB(color=green,thickness=3),mA(color=magenta,thickness=3),mA,mC(color=violet,thickness=3),O},printtext=true,axes=NONE);

> restart:with(geometry):

> _EnvHorizontalName:=x:_EnvVerticalName=y: The setting of the heights of the triangle points A, B, C and let's call it "Т"

> triangle(T,[point(A,7,8),point(B,6,-7),point(C,-6,7)]);

T

** Construction of the triangle**

> draw(T,axes=normal,view=[-8..8,-8..8]);

**The setting of the altitude in the triangle from the point Сaltitude(hC1,C,T,C1);**

> altitude(hC,C,T);

hC1

hC

** Analytical information about the altitude hC from the point С in the triangle**

> detail(hC);

name of the object: hC

form of the object: line2d

assume that the name of the horizonal and vertical axis are _x and _y

equation of the line: -99+_x+15*_y = 0

**Construction of the altitude from the point С**

> draw({T(color=gold,thickness=3),hC1(color=green,thickness=3),hC},printtext=true,axes=NONE);

** The setting of the altitude in the triangle from the point Аaltitude(hA1,A,T,A1);**

> altitude(hA,A,T);

hA1

hA

** Analytical information about the altitude hA from the point А in the triangle**

> detail(hA);

name of the object: hA

form of the object: line2d

assume that the name of the horizonal and vertical axis are _x and _y

equation of the line: -28-12*_x+14*_y = 0

** Construction of the altitude from the point А**

>draw({T(color=gold,thickness=3),hC1(color=green,thickness=3),hA1(color=red,thickness=3),hA1},printtext=true,axes=NONE); **The setting of the altitude from the point В**

> altitude(hB1,B,T,B1);

> altitude(hB,B,T);

hB1

hB

**Analytical information about the altitude hB from the point В in the triangle**

> detail(hB);

name of the object: hB

form of the object: line2d

assume that the name of the horizonal and vertical axis are _x and _y

equation of the line: -71+13*_x+_y = 0

** Consruction of the altitude from the point В**

>draw({T(color=gold,thickness=3),hC1(color=green,thickness=3),hA1(color=red,thickness=3),hB1(color=blue,thickness=3),hB1},printtext=true,axes=NONE);

**Calculation of the point of intersection of altitudes and let's call it "О"**

>intersection(O,hB,hA,hC);coordinates(O);

O

483 608

[---, ---]

97 97

** Construction of altitudes and marking of the point of intersection "О" in the triangle**

>draw({T(color=gold,thickness=3),hC1(color=green,thickness=3),hA1(color=red,thickness=3),hB1(color=blue,thickness=3),hB1,O},printtext=true,axes=NONE);