O POINT IS IN THE TRIANGLE ABC AND THE VECTOR OA 2OB 3OC 0 IS USED TO FIND THE RATIO OF THE AREA OF

Establish a coordinate system, a is (0,0), c is (x1,0), b is (x2,y2), o is (x,y), and the equation with the vector can be solved to get y2 = 3y

The area ratio is 3:1

OE OA+3OC( -2OB) b,o,e collinear,S AOC S(AODE) Think about it, why? ]

s（aoce）＝（1/2+1/6）s（aode）＝（2/3）s（aode），s（abco）＝s（aode）/3＝2s⊿aoc。

s⊿abc＝s（abco）+s⊿aoc＝3s⊿aoc。

The ratio of the triangle ABC to the AOC area is 3:1

Extend AO, pass C and B respectively as a triangle AOB, and the height of the triangle AOC is established, and take O as the origin to establish a Cartesian coordinate system.

Let a(0,a),b(x,y), c(a,b) vector oa+2ob+3oc=0

2x+3a=0

The area ratio of the triangle AOB is the absolute value of the triangle AOB of X B, and the area ratio of the triangle AOC is 3:2

Place the triangle abc into the coordinate system.

Let a=(2,0),b=(0,1).

Because the vector oa + 2ob + 4oc = 0

From the image. 4oc = 2 times the root number 2

modulo of vector oc = (root number 2) 2

Because oc is the angular bisector of the third quadrant.

So c(-1 2, -1 2).

The ratio of the area of OAB to OBC = high ratio = 2 (1 2) = 4:1

Vector oa+2ob+3oc=0, then there is.

Vector ao=2*vector ob+3*vector oc,(1)

Whereas, the vector ob = vector (ab-ao).(2)

Vector oc = vector (ac-ao).(3)

Substituting equations (2) and (3) into equations (1) obtains, the vector ao = vector (2 6*ab+3 6*ac)

Make ae=2 6ab on the ab side and af=3 6*ac. on the ac side

Then there is, vector ao = vector ae + vector af = ae + eo

Whereas, in the triangle aeo, ao sinaeo=eo sinbao=ao sinbac, eo*sinbac=aosinbao(4) formula.

s-abc=1/2*|ab|*|ac|*sinbac,s-bao=1/2*|ab|*|ao|*sinbao.

s-abc/s-bao=|ac|*sinbac/|ao|*sinbao.(5)

Substituting equation (4) into equation (5) yields s-abc s-bao=|ac|*sinbac/|eo|*sinbac

ac eo, while eo=af=3 6ac=1 2*ac

then, s-abc s-bao=ac eo=2 1

Then the ratio of the triangle ABC to the triangle AOC is 2:1

oa, ob, oc are all vectors, right?

The known conditions are respectively with oa, ob as a vector product of the forest, obtained.

2oa ob + 3oa oc = 0, ob oa + 3ob oc = 0, half of the modulus of the product to the filial piety ridge is equal to the area of the triangle this Shen pants, so there is.

2s(aob)=3s(aoc),s(aob)=3s(boc),s(abc):s(aoc)=3:1.

OE OA+3OC(-2OB) b,O,E collinear,S AOC S(AODE) 6s(abco) s(aoce) 2 [Think about it, why?] ]

s（aoce）＝（1/2+1/6）s（aode）＝（2/3）s（aode）， s（abco）＝s（aode）/3＝2s⊿aoc。

s⊿abc＝s（abco）+s⊿aoc＝3s⊿aoc。

The ratio of the triangle ABC to the AOC area is 3:1

You first draw a triangle, take the middle point of ab, take the inner point o, take the inner point o (in fact, the o point is on the cd), and connect it to oa ob oc od

oa+ob+2oc=0

2oC=oA+ob because the vector oa+ob=2od=-2oC (thus illustrating c o d three-point collinear).

Therefore o is the midpoint of cd (cd is the midline).

The ratio of the area of the ABC to the area of the triangle AOC is = 1:4

Let the intersection point of the line ao and the line bc be the point m, then.

Triangle obc and triangle abc area ratio =|om|：|am|。

Let om=xoa, then.

om=x(-2ob-3oc)。

and b, m, c collinear, we get -2x-3x=1. The solution yields x=-1 5.

Therefore, the area ratio of triangle obc and triangle ABC = 1 5:6 5 = 1:6.

So the area of the triangle obc is the triangle ob'12, while the area of the triangle ABO and the triangle ACO are both AB'c'The correct answer for 1 is: we take ob, then the ratio of the area of the triangle abc to the triangle obc is: (1, then o is the triangle ab'c'c'.

The point where OC is doubled is called B'12);the center of gravity; ,c';6;6+1 6+1 The area of 1 4 is also a triangle ab'c'Area of 1

Solution: OA+OC=-2OB

According to the parallelogram rule, the parallelogram OAEC, the parallelogram of OC OAEC, and OE intersect AC at the point D

Then oe=-2ob

So od=-ob

Both triangles are based on AC and have a ratio of 2:1 to height

So s(aoc):s(abc)=1:2

The answer is 1:2

Assuming that the trigonometric ABC is a regular triangle, the ratio of the area of AOC to the triangular ABC is 1:2