Show that every point in the interior of one circle is the orthocentre of another triangle inscribed in...
$begingroup$
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
geometric-construction
edited Feb 20 at 19:36
MarianD
2,1711618
asked Dec 24 '18 at 8:52
saisanjeevsaisanjeev
1,073312
$endgroup$
add a comment |
$begingroup$
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
geometric-construction
edited Feb 20 at 19:36
MarianD
2,1711618
asked Dec 24 '18 at 8:52
saisanjeevsaisanjeev
1,073312
$endgroup$
1
$begingroup$
This sound highly unlikely to be true.
$endgroup$
– Maria Mazur
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.
$endgroup$
– Maria Mazur
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.
$endgroup$
– Maria Mazur
Dec 24 '18 at 12:11
add a comment |
$begingroup$
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
geometric-construction
edited Feb 20 at 19:36
MarianD
2,1711618
asked Dec 24 '18 at 8:52
saisanjeevsaisanjeev
1,073312
$endgroup$
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
geometric-construction
geometric-construction
edited Feb 20 at 19:36
MarianD
2,1711618
asked Dec 24 '18 at 8:52
saisanjeevsaisanjeev
1,073312
edited Feb 20 at 19:36
MarianD
2,1711618
asked Dec 24 '18 at 8:52
saisanjeevsaisanjeev
1,073312
edited Feb 20 at 19:36
MarianD
2,1711618
edited Feb 20 at 19:36
MarianD
2,1711618
edited Feb 20 at 19:36
MarianD
2,1711618
2,1711618
asked Dec 24 '18 at 8:52
saisanjeevsaisanjeev
1,073312
asked Dec 24 '18 at 8:52
saisanjeevsaisanjeev
1,073312
asked Dec 24 '18 at 8:52
saisanjeevsaisanjeev
1,073312
1,073312
1
$begingroup$
This sound highly unlikely to be true.
$endgroup$
– Maria Mazur
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.
$endgroup$
– Maria Mazur
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.
$endgroup$
– Maria Mazur
Dec 24 '18 at 12:11
add a comment |
1
$begingroup$
This sound highly unlikely to be true.
$endgroup$
– Maria Mazur
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.
$endgroup$
– Maria Mazur
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.
$endgroup$
– Maria Mazur
Dec 24 '18 at 12:11
1
1
$begingroup$
This sound highly unlikely to be true.
$endgroup$
– Maria Mazur
Dec 24 '18 at 8:55
$begingroup$
This sound highly unlikely to be true.
$endgroup$
– Maria Mazur
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
$begingroup$
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.
$endgroup$
– Maria Mazur
Dec 24 '18 at 8:56
$begingroup$
I was talking about the problem.
$endgroup$
– Maria Mazur
Dec 24 '18 at 8:56
$begingroup$
oh. any way we can find such a point.
$endgroup$
– saisanjeev
Dec 24 '18 at 12:03
$begingroup$
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.
$endgroup$
– Maria Mazur
Dec 24 '18 at 12:11
$begingroup$
This can't be true, please read carefully your post again.
$endgroup$
– Maria Mazur
Dec 24 '18 at 12:11
add a comment |
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1
$begingroup$
This sound highly unlikely to be true.
$endgroup$
– Maria Mazur
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.
$endgroup$
– Maria Mazur
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.
$endgroup$
– Maria Mazur
Dec 24 '18 at 12:11