medians, heights, bisectors

[ins-]

In the acute angled triangle ABC - CP is the the foot of the perpendicular from C to AB. CL1 and CL2 are the angle bisectors of the angles ACP and BCP respectively (L1 and L2 are points from AB). M1 and M2 are the middles of CL1 and CL2. H1 and H2 are feets of perpendiculars from B to CL1 and A to CL2 respectively. M is the intersection point of BM1 and AM2. H is the intersection point of BH1 and AH2. Prove that P, H, M are colinear.

[Ilya Nek]

Let O1 is circumcenter of triangle CL1L2. Then
1. O1, H and P are colinear.
2. O1, M and H are colinear.
Do you need more details? or "art of problem solving" gave you full answer...

[ins-]

mathlinks is surprisingly silent about this problem ... there can be two cases - it is too easy or it is too hard.

[nnosipov]

ins- в сообщении #486443 писал(а):

In the acute angled triangle ABC

Clearly, your statement is true for arbitrary triangle $ABC$.

[ins-]

Even with the hint this problem is not easy. Can you give more details?
I said "acute angled" to make the problem a little easier.

[Ilya Nek]

Solution:
http://narod.ru/disk/26846611001/medians%2C.rar.html
It is too hard!

[nnosipov]

ins- в сообщении #487992 писал(а):

Can you give more details?

К сожалению, мои решения подобных задач имеют мало общего с геометрией --- это всего лишь вычисления с комплексными числами. Алгебра комплексных чисел равнодушна к тупоугольным треугольникам (равно как и к остроугольным), а вот геометрия, конечно, нет. Просто иногда бывает так, что вычисления оказываются не слишком громоздкими, поэтому их применение в некотором смысле оправдывается. Но в данной задаче это не так, здесь вручную не просчитать и необходим компьютер.

ins- в сообщении #487992 писал(а):

I said "acute angled" to make the problem a little easier.

Это понятно и разумно.

[ins-]

Thank you for the excellent solution. This problem as a geometric fact is not very beautiful, but I think it is good for some team selection test for imo or for some math magazine. What actualy make it valuable and beautiful are your opinions and the beautiful solution.