GEOMETRY MARATHON: SEASON 2
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I guess it's time to restart Geometry Marathon.
RULES: Very simple. I'll start this Marathon with a geometry problem. Next who can give the solution of the problem, will get a chance to post a new problem. Different solutions are welcome too. Who will give a different solution also can get a chance to post a new problem. To make things interesting, one have to post solution of a problem within 12 hours. After 12 hours, the problem will be a star problem and the problem will be re-posted in a new topic. But the Geometry Marathon will be continued by posting a new problem from the former solvers....
Hope u will enjoy the Geometry Marathon.
Problem 1:
Let $O$ be the circumcentre of an acute-angled triangle $ABC$. A line through $O$ intersects
the sides $CA$ and $CB$ at points $D$ and $E$ respectively, and meets the circumcircle of
triangle $ABO$ again at point $P$ (Not $O$) inside the triangle. A point $Q$ on side $AB$ is such
that $AQ/QB = DP/PE$
Prove that $\angle APQ = 2\angle CAP$
RULES: Very simple. I'll start this Marathon with a geometry problem. Next who can give the solution of the problem, will get a chance to post a new problem. Different solutions are welcome too. Who will give a different solution also can get a chance to post a new problem. To make things interesting, one have to post solution of a problem within 12 hours. After 12 hours, the problem will be a star problem and the problem will be re-posted in a new topic. But the Geometry Marathon will be continued by posting a new problem from the former solvers....
Hope u will enjoy the Geometry Marathon.
Problem 1:
Let $O$ be the circumcentre of an acute-angled triangle $ABC$. A line through $O$ intersects
the sides $CA$ and $CB$ at points $D$ and $E$ respectively, and meets the circumcircle of
triangle $ABO$ again at point $P$ (Not $O$) inside the triangle. A point $Q$ on side $AB$ is such
that $AQ/QB = DP/PE$
Prove that $\angle APQ = 2\angle CAP$
You spin my head right round right round,
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
Re: GEOMETRY MARATHON: SEASON 2
Is my figure correct?
Please Install $L^AT_EX$ fonts in your PC for better looking equations,
Learn how to write equations, and don't forget to read Forum Guide and Rules.
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes
Learn how to write equations, and don't forget to read Forum Guide and Rules.
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes
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- Posts:461
- Joined:Wed Dec 15, 2010 10:05 am
- Location:Dhaka
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Re: GEOMETRY MARATHON: SEASON 2
$Q$ is on $AB$
You spin my head right round right round,
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
- Abdul Muntakim Rafi
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Re: GEOMETRY MARATHON: SEASON 2
Hey guys,what software do you use to draw pictures(of course related to maths) in pc?
Man himself is the master of his fate...
Re: GEOMETRY MARATHON: SEASON 2
Please read Forum Guide and Rules before you post.
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
- FahimFerdous
- Posts:176
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- Location:Mymensingh, Bangladesh
Re: GEOMETRY MARATHON: SEASON 2
Hey, I've just started trying the problem. Don't post the solve yet please.
Actually, I'm a little busy these days.
Actually, I'm a little busy these days.
Your hot head might dominate your good heart!
Re: GEOMETRY MARATHON: SEASON 2
$\text{Solution(complex):}$
Last edited by *Mahi* on Sun Sep 18, 2011 9:26 am, edited 1 time in total.
Please read Forum Guide and Rules before you post.
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
- Abdul Muntakim Rafi
- Posts:173
- Joined:Tue Mar 29, 2011 10:07 pm
- Location:bangladesh,the earth,milkyway,local group.
Re: GEOMETRY MARATHON: SEASON 2
Thank you very much... @Mahi
Man himself is the master of his fate...
- Tahmid Hasan
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Re: GEOMETRY MARATHON: SEASON 2
i use cabriAbdul Muntakim Rafi wrote:Hey guys,what software do you use to draw pictures(of course related to maths) in pc?
বড় ভালবাসি তোমায়,মা
Re: GEOMETRY MARATHON: SEASON 2
New Problem!
Let $BE$ and $CF$ be the altitudes in an acute triangle $ABC$. Two circles passing through the points $A$ and $F$ are tangent to the line $BC$ at the points $P$ and $Q$ so that $B$ lies between $C$ and $Q$. Prove that the lines $PE$ and $QF$ intersect on the circumcircle of $\triangle AEF$.
(Intended to those who haven't solved it yet.)
Let $BE$ and $CF$ be the altitudes in an acute triangle $ABC$. Two circles passing through the points $A$ and $F$ are tangent to the line $BC$ at the points $P$ and $Q$ so that $B$ lies between $C$ and $Q$. Prove that the lines $PE$ and $QF$ intersect on the circumcircle of $\triangle AEF$.
(Intended to those who haven't solved it yet.)
Please read Forum Guide and Rules before you post.
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi