1. ## Triangle problem

Acute triangle is given. The angle at point C is 45 . Points A1 and B1 are the feet of the altitudes from points A and B, respectively . Let point H be an orthocenter of a triangle. On segments AA1 and BC and are points D and E, respectively, so that $\displaystyle A1D=A1E=A1B1$ . Prove that:

1. $\displaystyle CH=DE$

2.$\displaystyle A1B1^2=(A1B^2+A1C^2)/2$

2. ## Re: Triangle problem

Hi,
I think this problem has a fairly easy solution using analytic geometry. I've attached a diagram where the problem has been coordinatized; the diagram shows that the assumption that all angles of the triangle are acute is not necessary. I left most of the solution to you, but if you have question, just reply to this posting.

3. ## Re: Triangle problem

Thanks for the help johng ! But i was wondering how did you get coordinates of B1 (a+b/2, a-b/2) ?

4. ## Re: Triangle problem

Hi Matt1,
I've attached a slightly modified figure which shows how the coordinates for B1 are found. Again, reply if you have further questions.

5. ## Re: Triangle problem

Hi johng, again thanks for your fast reply and help, it's all clear now !