Question

prove that the area of an equilateral triangle divides on side of the squares equal to the half of the area of triangle described on one of its diagonals

Answer 1

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Answer 2

The three medians intersect at a single point, called the centroid of the triangle.
• Each median divides the triangle into two smaller triangles which have the same area.
• Because there are three vertices, there are of course three possible medians. 
• No matter what shape the triangle, all three always intersect at a single point. This point is called the centroid of the triangle.
• The three medians divide the triangle into six smaller triangles of equal area.
• The centroid (point where they meet) is the center of gravity of the triangle
• Two-thirds of the length of each median is between the vertex and the centroid, while one-third is between the centroid and the midpoint of the opposite side.
•

m=

2

b

2

+2

c

2

−

a

2

4

−

−

−

−

−

−

−

−

−

√

m=2b2+2c2−a24

, where

a

a

,

b

b

and

c

c

are the sides of the triangle and

a

a

is the side of the triangle whose midpoint is the extreme point of median

m

m

.

Area The number of square units it takes to exactly fill the interior of a triangle.

Usually called "half of base times height", the area of a triangle is given by the formula below.
•

A=

hb

2

A=hb2



Other formula:
•

A=

P∗r

2

A=P∗r2



•

A=

abc

4R

A=abc4R



Where

b

b

is the length of the base,

a

a

and

c

c

the other sides;

h

h

is the length of the corresponding altitude;

R

R

is the Radius of circumscribed circle;

r

r

is the radius of inscribed circle; P is the perimeter

• Heron's or Hero's Formula for calculating the area

A=

s(s−a)(s−b)(s−c)

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

√

A=s(s−a)(s−b)(s−c)

where

a,b,c

a,b,c

are the three sides of the triangle and

s=

a+b+c

2

s=a+b+c2

which is the semi perimeter of the triangle.

Perimeter The distance around the triangle. The sum of its sides. 

• For a given perimeter equilateral triangle has the largest area.
• For a given area equilateral triangle has the smallest perimeter.

Relationship of the Sides of a Triangle

• The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides. 

Interior angles The three angles on the inside of the triangle at each vertex. 

• The interior angles of a triangle always add up to 180°
• Because the interior angles always add to 180°, every angle must be less than 180°
• The bisectors of the three interior angles meet at a point, called the incenter, which is the center of the incircle of the triangle.

Exterior angles The angle between a side of a triangle and the extension of an adjacent side.


• An exterior angle of a triangle is equal to the sum of the opposite interior angles.
• If the equivalent angle is taken at each vertex, the exterior angles always add to 360° In fact, this is true for any convex polygon, not just triangles. 

Midsegment of a Triangle A line segment joining the midpoints of two sides of a triangle


• A triangle has 3 possible midsegments.
• The midsegment is always parallel to the third side of the triangle.
• The midsegment is always half the length of the third side. 
• A triangle has three possible midsegments, depending on which pair of sides is initially joined.

Relationship of sides to interior angles in a triangle

• The shortest side is always opposite the smallest interior angle
• The longest side is always opposite the largest interior angle

Angle bisector An angle bisector divides the angle into two angles with equal measures. 


• An angle only has one bisector.
• Each point of an angle bisector is equidistant from the sides of the angle.
• The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment DC is equal to the ratio of the length of side AB to the length of side