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 $C_1$. enter image description here 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

  • 1
    $\begingroup$ This sound highly unlikely to be true. $\endgroup$ – Aqua Dec 24 '18 at 8:55
  • $\begingroup$ any better ideas for such a construction. I would also like to know how to prove it wrong $\endgroup$ – saisanjeev Dec 24 '18 at 8:56
  • $\begingroup$ I was talking about the problem. $\endgroup$ – Aqua Dec 24 '18 at 8:56
  • $\begingroup$ oh. any way we can find such a point. $\endgroup$ – saisanjeev Dec 24 '18 at 12:03
  • $\begingroup$ This can't be true, please read carefully your post again. $\endgroup$ – Aqua Dec 24 '18 at 12:11

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