OliForum Matematico

Let ABC an acute triangle. The circle with diameter the altitude BD meets sides

AB,BC at K,L respectively. The tangents of this circle at K,L meet at M.Prove that

line BM bisects segment AC.

Babis

Have you any non metric solution to this problem ?

Of cource, it is easy to find that $ 3(c-a) = 2(m_c-m_a) $ .Then using the median formulae , we find a=c.

Μπάμπης

Excellent !!!

Babis

Well, a nice problem for beginers with olympiads is the following:

Problem

The incircle of a triangle ABC touches BC at D. Let E be the projection of B on the bisector of angle A. If M is the midpoint of side BC, prove that the triangle MDE is isosceles.

Enjoy it - Babis

Nice problem stergiu :wink: Let \alpha,\beta,\gamma be the angles \angle CAB,\angle ABC,\angle BCA . First case: M belongs to the arc AB wich not contains C. \angle MNA = \angle MBA because they lie on the same arc. Since M is on the bisector of the external angle B, \angle MBA = \dfrac{(\angle CAB...

After having seen and loged in at your wonderfull site, I put a simple geometry problem for students of class 10-12. . Problem The bisector of the internal angle A and the bisector of the external angle B of a triangle ABC meet the circumcircle of the triangle at points M,N.If I is the incenter of ...

Thank you very much ! I visited these links.

To be more acurate, I was looking for the problem for year 2009, but nobody has them. Probably the competition did not appear this year.
If I learn something new, I will let you know !

Regards - Babis

Hallo to all math friends in italy ! I'd like to ask you if you know the problems and solutions for maditerranean olympiads (years 2000 - 2009), or a link(in italian, english, spanish ect). I was asked for the problems by a greek student who prepares himself for IMO , but at moment I can not be in c...