Geometry Marathon : Season 3
problem 55 :
Let$ ABCD $be a cyclic quadrilateral, let the tangents to$ A$ and $C$ intersect
in $P$, the tangents to$ B $and$ D$ in$ Q$. Let$ R$ be the intersection of$ AB $and$ CD $and let
$S $be the intersection of$ AD$ and$ BC$. Show that$ P,,R, S $are collinear.
Let$ ABCD $be a cyclic quadrilateral, let the tangents to$ A$ and $C$ intersect
in $P$, the tangents to$ B $and$ D$ in$ Q$. Let$ R$ be the intersection of$ AB $and$ CD $and let
$S $be the intersection of$ AD$ and$ BC$. Show that$ P,,R, S $are collinear.
Re: Geometry Marathon : Season 3
$\textbf{Solution 55}$Dustan wrote: ↑Sun Mar 14, 2021 7:22 pmproblem 55 :
Let$ ABCD $be a cyclic quadrilateral, let the tangents to$ A$ and $C$ intersect
in $P$, the tangents to$ B $and$ D$ in$ Q$. Let$ R$ be the intersection of$ AB $and$ CD $and let
$S $be the intersection of$ AD$ and$ BC$. Show that$ P,,R, S $are collinear.
Re: Geometry Marathon : Season 3
$\textbf{Problem 56}$
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$.
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$.
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Re: Geometry Marathon : Season 3
Click "Full editor and preview " then "attachment "
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Re: Geometry Marathon : Season 3
Solution(It would be kind if somebody confirm it ):~Aurn0b~ wrote: ↑Sun Mar 14, 2021 11:12 pm$\textbf{Problem 56}$
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$.
Hmm..Hammer...Treat everything as nail
Re: Geometry Marathon : Season 3
i guess right.post the next one.
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Re: Geometry Marathon : Season 3
Thanks.But currently i ran out of problems so you may post one which can be solved without projec or inversive geo....
Hmm..Hammer...Treat everything as nail
- Anindya Biswas
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Problem 57
Does there exists $6$ points in the plane such that the distance between any two of them is an integer?
Edit : I forgot to mention that no $3$ points collinear.
Source :
Edit : I forgot to mention that no $3$ points collinear.
Source :
Last edited by Anindya Biswas on Sun Mar 21, 2021 11:22 pm, edited 1 time in total.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
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Re: Problem 57
I think we can draw 6 points that have an integer distance from any 2, if all of them are on the same line.Anindya Biswas wrote: ↑Sun Mar 21, 2021 9:02 pmDoes there exists $6$ points in the plane such that the distance between any two of them is an integer?
Source :
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré
-Henri Poincaré