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December 7th, 2017, 02:34 AM  #1 
Newbie Joined: Dec 2017 From: Spain Posts: 5 Thanks: 0  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 04:28 AM. 
December 7th, 2017, 05:43 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,248 Thanks: 1439 
Do you have a diagram?

December 7th, 2017, 05:45 AM  #3 
Newbie Joined: Dec 2017 From: Spain Posts: 5 Thanks: 0 
how a diagram?

December 7th, 2017, 06:09 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,248 Thanks: 1439 
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, 08:12 AM  #5 
Newbie Joined: Dec 2017 From: Spain Posts: 5 Thanks: 0 
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 09:43 AM. 
December 7th, 2017, 09:45 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 18,248 Thanks: 1439 
Can you upload the image to this forum?

December 8th, 2017, 02:10 AM  #7 
Newbie Joined: Dec 2017 From: Spain Posts: 5 Thanks: 0 
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, 07:47 AM  #8 
Member Joined: Jan 2016 From: Athens, OH Posts: 61 Thanks: 37 
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 07:52 AM. 
December 9th, 2017, 01:53 AM  #9 
Newbie Joined: Dec 2017 From: Spain Posts: 5 Thanks: 0 
Don't worry guys, I already solved it. Thanks for all. Last edited by skipjack; December 9th, 2017 at 05:18 AM. 
December 9th, 2017, 05:24 AM  #10 
Global Moderator Joined: Dec 2006 Posts: 18,248 Thanks: 1439 
For the benefit of all those who viewed this problem, can you post your solution?


Tags 
circles, cyclic hexagon, exercise, geometry, involved, orthocenter, parallels 
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