143. Assess of a ratio (nice and difficult problem).
by Virgil Nicula, Oct 5, 2010, 3:12 PM
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=153668
Lemma. Let
be a triangle and let 
be a point for which 
. The second tangent from the point 
to the incircle of
touches it at the point 
. Denote the intersection 
. Prove that 
.
Remark. The proposed problem and the its Pohoatza"s generalization from
http://www.artofproblemsolving.com/viewtopic.php?t=152994 prove easily (even immediately) with the above lemma.
Proof. Denote the incircle
of the triangle and the tangent points 
. Suppose w.l.o.g. 
and denote 
and observe that
.Thus, 
.
Apply the Brianchon's theorem to the circumscriptible quadrilateral
i.e. there is 
.
Apply the Menelaus' theorem to the transversal 
in 
:
.
Apply the Menelaus' theorem to the transversal 
in :
.
Observe that
.
Apply the Menelaus' theorem to the transversal 
in 
:
![$\boxed{\frac{XM}{m\left[m(p-c)-(p-b)\right]}=\frac{XC}{(1+m)(p-b)}=\frac{a}{(1+m)\left[m^{2}(p-c)+(p-b)\right]}}\ \ (5)$](//latex.artofproblemsolving.com/2/6/c/26c8fa754ca54285b1d337765a9cbfbe8e18969e.png)
.
Therefore,
.
Lemma. Let
be a triangle and let 
be a point for which 
. The second tangent from the point 
to the incircle of
touches it at the point 
. Denote the intersection 
. Prove that 
.Remark. The proposed problem and the its Pohoatza"s generalization from
http://www.artofproblemsolving.com/viewtopic.php?t=152994 prove easily (even immediately) with the above lemma.
Proof. Denote the incircle
of the triangle and the tangent points 
. Suppose w.l.o.g. 
and denote 
and observe that
.Thus, 
.Apply the Brianchon's theorem to the circumscriptible quadrilateral
i.e. there is 
.
Apply the Menelaus' theorem to the transversal 
in 
:
.
Apply the Menelaus' theorem to the transversal 
in :
.Observe that
.
Apply the Menelaus' theorem to the transversal 
in 
:
![$\boxed{\frac{XM}{m\left[m(p-c)-(p-b)\right]}=\frac{XC}{(1+m)(p-b)}=\frac{a}{(1+m)\left[m^{2}(p-c)+(p-b)\right]}}\ \ (5)$](http://latex.artofproblemsolving.com/2/6/c/26c8fa754ca54285b1d337765a9cbfbe8e18969e.png)
.Therefore,
.This post has been edited 8 times. Last edited by Virgil Nicula, Dec 1, 2015, 11:04 AM
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