Distance Euler's Line

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petarantes

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In a triangle ABC, H, G, and O are orthocenter, centroid, and circumcenter of the triangle. If the Euler's line is parallel AC and m <(HBC) = 2m <(OCA), calculate GO if AH = a

I tried to draw the triangle and relate the properties but couldn't find a solution. We know that GH = 2GO and BG = 2GP Triangle BHG ~ POG
 

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Please show us your work so we can find errors and/or give suggestions.
 
There's quite a lot of work visible in the picture! But a couple things may help others help you.

First, putting it in words can help us follow your thinking and see where it might be extended; second, if this comes from a context in which you have learned some particular theorems, stating those might give us a hint or two. If it's just a random contest-type problem, then we'll just have to work with you on thinking it through.

The first thing I think I'd do (I haven't yet) is to think about the implications of the mere fact that Euler's line is parallel to AC. How does that restrict what triangles we can be starting with?
 
Dr.Peterson said:
There's quite a lot of work visible in the picture! But a couple things may help others help you.

First, putting it in words can help us follow your thinking and see where it might be extended; second, if this comes from a context in which you have learned some particular theorems, stating those might give us a hint or two. If it's just a random contest-type problem, then we'll just have to work with you on thinking it through.

The first thing I think I'd do (I haven't yet) is to think about the implications of the mere fact that Euler's line is parallel to AC. How does that restrict what triangles we can be starting with?
Click to expand...
Its a random context-type. I think if O is orthocenter, OAC is an isosceles triangle. (OA = OC = radius circumscribe circumference)
By the property that Euler's line HG = 2GO
The solution of the problem is GO = a/3 therefore HO would need to be equal a "a" and so AHO
should be an isosceles triangle but I couldn't demonstrate that AHO is isosceles.
 
I had determined that the answer appeared to be a/3, by making a drawing on GeoGebra, just dragging B around until the conditions were (approximately) met. I'll keep looking as I have time, and hope others will, too!
 
I found an answer.

Extend AH, and use that to find angle CAH. Then combine that with what you know about angle CAO.
 
Thanks a lot for the help.

I managed to solve the problem and the answer is really a/3
 
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