110. German TST 2004 - Exam V , Problem 1
by Virgil Nicula, Sep 11, 2010, 10:25 PM
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=50&t=155550&p=874329#p874329
Proof I. Suppose w.l.o.g. that
. Denote 
. Observe that : 
and
.
Apply the Menelaus' theorem to the 
.
Apply the Menelaus' theorem to 
.
From
, 
results that 
i.e. the lines 
, 
are concurrently in the point 
for which 
.
Lemma. Let
for which 
. For a mobile line 
for which 
denote the points 
.
Define the point
for which 
and 
. Then 
and 
, where 
.
Proof. Indeed,
and 
, i.e. .
Otherwise. Denote 
.
Therefore,
.
Proof II. Apply the above lemma to the right triangle
and the line 
.
Quote:
The 
- excircle of a triangle 
touches the side 
at the point 
and the extended side 
at the point 
. The 
- excircle
touches the lines
and at the points 
and 
respectively. Denote 
. Show that 
.
- excircle of a triangle 
touches the side 
at the point 
and the extended side 
at the point 
. The 
- excircletouches the lines
and at the points 
and 
respectively. Denote 
. Show that 
.Proof I. Suppose w.l.o.g. that
. Denote 
. Observe that : 
and
.
Apply the Menelaus' theorem to the 
.
Apply the Menelaus' theorem to 
.From
, 
results that 
i.e. the lines 
, 
are concurrently in the point 
for which 
.Lemma. Let
for which 
. For a mobile line 
for which 
denote the points 
.Define the point
for which 
and 
. Then 
and 
, where 
.Proof. Indeed,
and 
, i.e. .
Otherwise. Denote 
.Therefore,
.Proof II. Apply the above lemma to the right triangle
and the line 
.This post has been edited 15 times. Last edited by Virgil Nicula, Nov 23, 2015, 8:08 AM
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