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December 7th, 2017, 02:34 AM   #1
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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.
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December 7th, 2017, 05:43 AM   #2
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Do you have a diagram?
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December 7th, 2017, 05:45 AM   #3
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how a diagram?
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December 7th, 2017, 06:09 AM   #4
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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.
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December 7th, 2017, 08:12 AM   #5
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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.
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December 7th, 2017, 09:45 AM   #6
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Can you upload the image to this forum?
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December 8th, 2017, 02:10 AM   #7
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Let me explain what the colors mean:
Blue: Original triangle
Green: Altitudes
Purple: Resultant circles
Red: circle with the common lying points
Attached Images
File Type: jpg Captura de pantalla 2017-12-08 a las 11.06.48.jpg (20.5 KB, 4 views)
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December 8th, 2017, 07:47 AM   #8
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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.
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December 9th, 2017, 01:53 AM   #9
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Don't worry guys, I already solved it.
Thanks for all.

Last edited by skipjack; December 9th, 2017 at 05:18 AM.
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December 9th, 2017, 05:24 AM   #10
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For the benefit of all those who viewed this problem, can you post your solution?
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circles, cyclic hexagon, exercise, geometry, involved, orthocenter, parallels



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