# Triangle problem

• Jun 24th 2013, 11:12 AM
matt1
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\$
• Jun 25th 2013, 06:00 AM
johng
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.

Attachment 28668
• Jun 25th 2013, 01:21 PM
matt1
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) ?
• Jun 25th 2013, 04:45 PM
johng
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.

Attachment 28669
• Jun 26th 2013, 08:46 AM
matt1
Re: Triangle problem
Hi johng, again thanks for your fast reply and help, it's all clear now !