BdMO Online Forum

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 $\angle APQ = 2\angle CAP$

Is my figure correct?

$Q$ is on $AB$

Hey guys,what software do you use to draw pictures(of course related to maths) in pc?

Hey, I've just started trying the problem. Don't post the solve yet please.
Actually, I'm a little busy these days.

$\text{Solution(complex):}$

Thank you very much... @Mahi

Abdul Muntakim Rafi wrote:Hey guys,what software do you use to draw pictures(of course related to maths) in pc?

i use cabri

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.)