Geometry

**simplependulum** · July 16th 2010, 10:01 PM

Let $P$ be a point inside the triangle $ABC$ . The lines $AP , BP$ and $CP$ intersect the circumcircle $\Gamma$ of triangle $ABC$ agian at the points $K,L$ and $M$ respectively . The tagent to $\Gamma$ at $C$ intersects the line $AB$ at $S$ . Suppose that $SC = SP$ . Prove that $MK=ML$ .

If you could answer this problem , you are able to receive the Honourable Mention of this year IMO .

Moderator edit: Approved Challenge question.

**Chris11** · July 17th 2010, 10:23 AM

Did you recieve the honourable mention for this year's IMO?

**chiph588@** · July 17th 2010, 03:19 PM

Originally Posted by simplependulum

Let $P$ be a point inside the triangle $ABC$ . The lines $AP , BP$ and $CP$ intersect the circumcircle $\Gamma$ of triangle $ABC$ agian at the points $K,L$ and $M$ respectively . The tagent to $\Gamma$ at $C$ intersects the line $AB$ at $S$ . Suppose that $SC = SP$ . Prove that $MK=ML$ .

If you could answer this problem , you are able to receive the Honourable Mention of this year IMO .

May be a dumb question, but what exactly is $\Gamma$ in this problem?

**tonio** · July 17th 2010, 06:43 PM

Originally Posted by chiph588@

May be a dumb question, but what exactly is $\Gamma$ in this problem?

$\Gamma$ is the circumcircle = the unique circle passing through all the three vertices of the triangle.

Tonio

**chiph588@** · July 17th 2010, 07:46 PM

Originally Posted by tonio

$\Gamma$ is the circumcircle = the unique circle passing through all the three vertices of the triangle.

Tonio

Wow, I originally read circumference instead of circumcircle!

**simplependulum** · July 18th 2010, 09:15 PM

I don't know , only the contestants getting full mark of at least one problem (of 6 ) would receive the honourable mention in IMO . I am not powerful enough to take a seat in IMO ( hope it's just now but not in the furture ) . My solution looks fine to me but i am not confident to get full mark !

**Wilmer** · July 21st 2010, 05:33 AM

Your question is complete confusion to me...but in case this helps:

Circle radius 65 is circumcircle of triangle ABC: BC=78, AC=120, AB = 126.
Point P is inside triangle ABC: PA=104, PB=50, PC=32.
In other words: a completely integer case.
This finding of mine not enough for an IMO; can I get just the IM ?

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