Geometry

• Jul 16th 2010, 10:01 PM
simplependulum
Geometry
Let \$\displaystyle P \$ be a point inside the triangle \$\displaystyle ABC \$ . The lines \$\displaystyle AP , BP \$ and \$\displaystyle CP \$ intersect the circumcircle \$\displaystyle \Gamma\$ of triangle \$\displaystyle ABC\$ agian at the points \$\displaystyle K,L\$ and \$\displaystyle M\$ respectively . The tagent to \$\displaystyle \Gamma\$ at \$\displaystyle C\$ intersects the line \$\displaystyle AB\$ at \$\displaystyle S\$ . Suppose that \$\displaystyle SC = SP\$ . Prove that \$\displaystyle MK=ML\$ .

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

Moderator edit: Approved Challenge question.
• Jul 17th 2010, 10:23 AM
Chris11
Did you recieve the honourable mention for this year's IMO?
• Jul 17th 2010, 03:19 PM
chiph588@
Quote:

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

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

May be a dumb question, but what exactly is \$\displaystyle \Gamma \$ in this problem?
• Jul 17th 2010, 06:43 PM
tonio
Quote:

Originally Posted by chiph588@
May be a dumb question, but what exactly is \$\displaystyle \Gamma \$ in this problem?

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

Tonio
• Jul 17th 2010, 07:46 PM
chiph588@
Quote:

Originally Posted by tonio
\$\displaystyle \Gamma\$ is the circumcircle = the unique circle passing through all the three vertices of the triangle.

Tonio

• Jul 18th 2010, 09:15 PM
simplependulum
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 !
• Jul 21st 2010, 05:33 AM
Wilmer
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 ?