December 7th, 2017, 01:34 AM  #1 
Newbie Joined: Dec 2017 From: Spain Posts: 15 Thanks: 1  Geometry exercise where an orthocenter and parallels are involved
Good Morning mathslovers I've recently received a geometry exercise that I'm not able to solve. It goes so: In the acute triangle ABC, the orthocenter (cut of the altitudes) is defined by H. The altitude of A cuts the side BC in H(a) and the parallel line of BC through H cuts the circle with diameter AH(a) in the points P(a) and Q(a). You'll find the points P(b) and Q(b) as P(c) and Q(c) analogously. Prove that the six points P(a); Q(a); P(b);Q(b); P(c); Q(c) lie on a common circle. I've tried to solve the problem with analytic geometry (with coordinates and so) but I got to big terms so I just gave up. Please have a look and post all ideas you have to solve it (it doesn't matter if it isn't the complete answer, a hint would also help). Thanks for all. Last edited by skipjack; December 7th, 2017 at 03:28 AM. 
December 7th, 2017, 04:43 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,962 Thanks: 1606 
Do you have a diagram?

December 7th, 2017, 04:45 AM  #3 
Newbie Joined: Dec 2017 From: Spain Posts: 15 Thanks: 1 
how a diagram?

December 7th, 2017, 05:09 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,962 Thanks: 1606 
If you don't have a diagram, draw a large acuteangled triangle, then its altitudes, then the parallel lines defined, etc. It's rather tedious to make the drawing, but a good diagram tends to help.

December 7th, 2017, 07:12 AM  #5 
Newbie Joined: Dec 2017 From: Spain Posts: 15 Thanks: 1 
Drawing it in a diagram didn't really help, due to the fact that there were too many lines involved. Any other ideas?? Here is the link to the image file:///Users/miguelvaldiviesovalles/Desktop/Captura%20de%20pantalla%2020171207%20a%20las%2017.08.38.png Last edited by skipjack; December 7th, 2017 at 08:43 AM. 
December 7th, 2017, 08:45 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 18,962 Thanks: 1606 
Can you upload the image to this forum?

December 8th, 2017, 01:10 AM  #7 
Newbie Joined: Dec 2017 From: Spain Posts: 15 Thanks: 1 
Let me explain what the colors mean: Blue: Original triangle Green: Altitudes Purple: Resultant circles Red: circle with the common lying points 
December 8th, 2017, 06:47 AM  #8 
Member Joined: Jan 2016 From: Athens, OH Posts: 88 Thanks: 47 
This problem is solved (2 solutions) at cut_the_knot  https://www.cuttheknot.org/Curricu...shtml#solution Oops. Similar problem but different. Last edited by johng40; December 8th, 2017 at 06:52 AM. 
December 9th, 2017, 12:53 AM  #9 
Newbie Joined: Dec 2017 From: Spain Posts: 15 Thanks: 1 
Don't worry guys, I already solved it. Thanks for all. Last edited by skipjack; December 9th, 2017 at 04:18 AM. 
December 9th, 2017, 04:24 AM  #10 
Global Moderator Joined: Dec 2006 Posts: 18,962 Thanks: 1606 
For the benefit of all those who viewed this problem, can you post your solution?


Tags 
circles, cyclic hexagon, exercise, geometry, involved, orthocenter, parallels 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Geometry exercise  greg1313  Geometry  1  September 24th, 2015 10:30 PM 
Is any property of orthocenter involved here ?  saravananr  Geometry  1  November 14th, 2014 02:41 AM 
Algebraic Geometry  D&F Section 15.1, Exercise 24  Math Amateur  Abstract Algebra  0  October 30th, 2013 07:36 PM 
Incenter and Orthocenter  Albert.Teng  Algebra  0  January 21st, 2013 05:06 PM 
Parallels of the golden section and the Eulerian number  PerAA  Number Theory  1  November 25th, 2012 04:21 AM 