296. Some nice Vittasko's problems.
by Virgil Nicula, Jul 11, 2011, 3:31 AM
PP1. Let 
be a triangle and let 
be its altitude, where 
. Draw a circle with the diameter ![$[AD]$](//latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
which intersects the sides 
, 
at
the points
, 
respectively and again the circumcircle 
of 
at the point 
. Prove that the lines 
, 
and 
are concurrently.
Proof 1 (metric) of PP1. Denote
, the diameter ![$[AA']$](//latex.artofproblemsolving.com/f/8/3/f83bdc8e170d3bc867097c603fa03cf6edefbb4b.png)
of 
and 
, 
, 
. Suppose w.l.o.g. that 
, i.e.
. Observe that 
and since the rays 
, 
are the 
-symmedians in 
, 
espectively obtain that 
and
. Apply the Ceva's theorem to the point 
and the triangle : 
. Thus, 
and
. Apply the Menelaus' theorem to the
transversal
and 
. Denote 
and
. Observe that 
, i.e. the quadrilateral 
is cyclically 
. Denote 
and
. Observe that 
. Apply the Sinus' theorem in 
![$2R\cdot \sin[\phi -(B-C)]=$](//latex.artofproblemsolving.com/a/5/3/a53aaac448d0490dc4b305b747e61b13fbe01616.png)
. From the well-known property 
obtain that 
. In conclusion, the lines , and are concurrently the lines 
and 
are concurrently
, what is truly. I used the identity 
.
Lemma. Denote in the points 
and 
. Then 
.
Proof 2 (synthetic - Jayme) of PP1. Denote the diameter of . Prove easily that and observe that 
, 
and . Therefore, 
is homothetically with 
, i.e. 
.
be a triangle and let 
be its altitude, where 
. Draw a circle with the diameter ![$[AD]$](http://latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
which intersects the sides 
, 
atthe points
, 
respectively and again the circumcircle 
of 
at the point 
. Prove that the lines 
, 
and 
are concurrently.Proof 1 (metric) of PP1. Denote
, the diameter ![$[AA']$](http://latex.artofproblemsolving.com/f/8/3/f83bdc8e170d3bc867097c603fa03cf6edefbb4b.png)
of 
and 
, 
, 
. Suppose w.l.o.g. that 
, i.e.
. Observe that 
and since the rays 
, 
are the 
-symmedians in 
, 
espectively obtain that 
and
. Apply the Ceva's theorem to the point 
and the triangle : 
. Thus, 
and
. Apply the Menelaus' theorem to thetransversal
and 
. Denote 
and
. Observe that 
, i.e. the quadrilateral 
is cyclically 
. Denote 
and
. Observe that 
. Apply the Sinus' theorem in 
![$2R\cdot \sin[\phi -(B-C)]=$](http://latex.artofproblemsolving.com/a/5/3/a53aaac448d0490dc4b305b747e61b13fbe01616.png)
. From the well-known property 
obtain that 
. In conclusion, the lines , and are concurrently the lines 
and 
are concurrently 
, what is truly. I used the identity 
. Lemma. Denote in
the points 
and 
. Then 
.
, 
, 
. Apply the van Aubel's relation 
. Apply the Menelaus' theorem to the transversal 
and 
(a symmetrical expression in 
, 
)
, i.e. 
for which 
and 
m the points 
, 
and
. Therefore, 
.
, what is truly. In conclusion, 
, i.e. 
and 
are cyclically 
.
and 
. Obtain that 
.Proof 2 (synthetic - Jayme) of PP1. Denote the diameter
of . Prove easily that and observe that 
, 
and
. Therefore, 
is homothetically with 
, i.e. 
. This post has been edited 68 times. Last edited by Virgil Nicula, Nov 21, 2015, 7:53 AM
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