Vector method in geometry

[haduyhung]

Dear friends,

I am preparing some geometry problems in which we could using
vector method to solve them ie... Really i got about 30 simple problems and very few nice problems or hard problems[Special i do not have much such problems which appear in Olympiads].If you have any exprience about this please let me know!
Thanks in advanced!
Yours,

[beata]

Here are two problems:

1. Prove that if in convex quadrilateral ABCD holds the equality
|AB|^2 + |CD|^2 = |BC|^2 + |DA|^2
then AC and BD are perpendicular.

2. Given a triangle ABC, over the sides AC and BC we construct squares. Denote the centres of these squares K and L, and D the centre of AB. Prove that the triangle KDL is right-angled and isosceles.

[Lagrangia]

here is a link where you may find some problems like that :
http://popacalin.tripod.com/calinpopa/geometry.pdf

[amfulger]

1. Let A1, A2, A3, A4 be distinct points on a circle. Let Wi be Euler's nine points circle of triangle AjAkAl where {i,j,k,l}={1,2,3,4}. Prove that this circles meet at a point.
(you don't realy nead the 4 points to be on a circle, but you do need them on a circle if you want a nice vector solution).

2. It's odds on you know this problem:
Let A,B,C,D be distinct points on a circle centered on O, so that AC and BD are perpendicular, and let T=AC \cap BD.
Prove that 2*OT= OA+OB+OC+OD (vectors not segments).

[Arne]

I am not fond of vectors... I don't think you can apply it very often. (Maybe it's a good idea to master complex numbers instead: they are more powerful than vectors but not much harder to work with.)

[Ashegh]

vecters are not so power full . but in some problems if u use them ,theproblems will solved very nice and soon. .

[Ashegh]

for example:three trisngles ,in wich each one has equal sides(OAB,OCD,OEF).
X is the midpoint of FA.Y is the midpoint of BC.and Z is the mid point of ED.
prove that :triangle XYZ has equal sides.


try to solve it by using vectors .

[Arne]

Yes, but it's even easier to prove that using complex numbers, so I think it is better to master complex numbers, rather than vectors.

[Ashegh]

u are right Arne .

[Phelpedo]

But what about solid geometry?

[N.T.TUAN]

[Fabiomr1]

[beyrem]

well i don't like solving geometry problems with vectors or complex numbers.....
but if you want, the first problem of the last imo con be solved with vectors..
it's a good example... i didn't solved it like that in spain but it could be done with simple vectors' manipulations ..

[apratimdefermat]

Can anyone recommend a good book which teaches how to use vectors in geometry with a lot of Olympiad standard examples and exercises.


I think barycentrics are also a very good technique in solving many problems

[DIMITRIS T]

[jayme]

Dear Mathlinkers,
I think that the tool "vector" can be used for solving every geometric problem involving lines and circles.
The knot is that the relevant technics are not clearly teach and named like the most important theorems in Geometry.
In my honest point of view, this space is very open for those who want to developp a personnal approach. I have investigated one time the marvellous space...
Sincerely
Jean-Louis