# Show that every point in the interior of one circle is the orthocentre of another triangle inscribed in another circle

Let $$C_1$$ and $$C_2$$ be two circles in the plane with radius R and 3R respectively. Show that every point in the interior of $$C_2$$ is the orthocentre of some triangle inscribed in $$C_1$$. I gave a construction as follows. Take any point, call it H in the interior of $$C_2$$. Join OH, it intersects the circle $$C_1$$ at two points say $$A$$ and $$X$$ with $$A$$ being nearer to $$H$$. Construct perpendicular bisector of $$AX$$. Let it intersect $$C_1$$ at $$B$$ and $$C$$. I tell that $$ABC$$ is the required triangle.

If I assume $$H$$ to be the orthocentre then all the properties are matching. However, I am unable to prove that the above construction will guarantee that H will be the orthocentre of triangle ABC.

Any help will be appreciated. Thanks in advance

• This sound highly unlikely to be true. – Aqua Dec 24 '18 at 8:55
• any better ideas for such a construction. I would also like to know how to prove it wrong – saisanjeev Dec 24 '18 at 8:56
• I was talking about the problem. – Aqua Dec 24 '18 at 8:56
• oh. any way we can find such a point. – saisanjeev Dec 24 '18 at 12:03
• This can't be true, please read carefully your post again. – Aqua Dec 24 '18 at 12:11