My Math Forum Geometry exercise where an orthocenter and parallels are involved

 Geometry Geometry Math Forum

 December 7th, 2017, 02:34 AM #1 Newbie   Joined: Dec 2017 From: Spain Posts: 18 Thanks: 1 Geometry exercise where an orthocenter and parallels are involved Good Morning maths-lovers 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: 20,110 Thanks: 1909 Do you have a diagram?
 December 7th, 2017, 05:45 AM #3 Newbie   Joined: Dec 2017 From: Spain Posts: 18 Thanks: 1 how a diagram?
 December 7th, 2017, 06:09 AM #4 Global Moderator   Joined: Dec 2006 Posts: 20,110 Thanks: 1909 If you don't have a diagram, draw a large acute-angled 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. Thanks from MIKI14
 December 7th, 2017, 08:12 AM #5 Newbie   Joined: Dec 2017 From: Spain Posts: 18 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%202017-12-07%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: 20,110 Thanks: 1909 Can you upload the image to this forum?
December 8th, 2017, 02:10 AM   #7
Newbie

Joined: Dec 2017
From: Spain

Posts: 18
Thanks: 1

Let me explain what the colors mean:
Blue: Original triangle
Green: Altitudes
Purple: Resultant circles
Red: circle with the common lying points
Attached Images
 Captura de pantalla 2017-12-08 a las 11.06.48.jpg (20.5 KB, 6 views)

 December 8th, 2017, 07:47 AM #8 Member   Joined: Jan 2016 From: Athens, OH Posts: 92 Thanks: 47 This problem is solved (2 solutions) at cut_the_knot -- https://www.cut-the-knot.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: 18 Thanks: 1 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: 20,110 Thanks: 1909 For the benefit of all those who viewed this problem, can you post your solution?

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post greg1313 Geometry 1 September 24th, 2015 11:30 PM saravananr Geometry 1 November 14th, 2014 03:41 AM Math Amateur Abstract Algebra 0 October 30th, 2013 08:36 PM Albert.Teng Algebra 0 January 21st, 2013 06:06 PM PerAA Number Theory 1 November 25th, 2012 05:21 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top