149. Some (8) problems with collinearity or concurrence.
by Virgil Nicula, Oct 7, 2010, 8:42 PM
PP1. Denote the tangent 
to the circle 
in the point 
. Let 
be a triangle with the circumcircle . For an interior point 
w.r.t. 
denote
the second intersections
, 
, 
of 
, 
, 
with the circle and 
, 
, 
. Prove that 
.
Proof. Denote
. From the well-known relations 
and 
obtain
. Show analogously 
and 
. Apply the Menelaus' theorem :
because 
. In conclusion, .
PP2 (Матматика в школе, nr.4/1984, pr. 2760). Let with circumcirle 
. Denote 
. Prove that .
Proof. Denote the orthocenter
of and 
. Observe that the circumcircle of 
is the circle with diameter ![$[OH]$](//latex.artofproblemsolving.com/7/b/8/7b812c672802ddc509741f60b6fd241be97668af.png)
and
.Therefore, the points 
, 
, 
are inversely to the points 
, 
, 
respectively in the inversion
with the modulus
and the pole 
which belongs to the circumcircle of . In conclusion, and 
.
PP3. Let
be a hexagon which is inscribed in a circle . For a point 
define the points 
,
, 
. Prove that 
.
Proof. Apply the Pascal's theorem to the following cyclical hexagons :
.
PP4. Consider
, 
so that 
and 
. For 
define 
, 
,
, 
. Prove that 
, 
, 
are concurrently in for which 
and 
(V.N.).
Proof. Denote
, 
. Apply the Menelaus' theorem to the transversals :
.
.
. From the relation 
obtain 
.
Thus,
. From 
obtain 
and from 
obtain 
, i.e. 
. In conclusion, 
.
Remark. Can prove that the points
coincides with only with the relation which is independently by the relation . Indeed, apply the Ceva's' theorem to :
. From the definition of the point
obtain
. Therefore, 
, i.e. 
. The position of the fixed point 
is independently by the choosing of the points
, for which 
and the slope of the line 
, i.e. this problem has three degrees of freedom.
PP5. Let be a triangle. Consider the pairs of isotomic points ![$\{D,D'\}\subset [BC]$](//latex.artofproblemsolving.com/9/6/8/968e5a9b8adc151035f8b67c7b1ceb7a8b72ca8d.png)
, ![$\{E,E'\}\subset [CA]$](//latex.artofproblemsolving.com/7/f/5/7f57713459b458e4e59d6945e2cb05d3031898cd.png)
, ![$\{F,F'\}\subset [AB]$](//latex.artofproblemsolving.com/a/b/b/abb989c121c9b1d08c08621abe9016a64400fbf3.png)
. Prove that
the parallels through the points 
, 
, 
to the lines 
, 
, 
respectively are concurrently (V.N.).
Proof.
PP6. Let be a triangle with the incircle 
which touches its sides in 
, 
, 
. Denote the midpoints , , of the sides
![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, ![$[CA]$](//latex.artofproblemsolving.com/4/5/c/45c1acd47628de406680d04c09fe6314c3847acf.png)
, ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
respectively. Prove that the lines which pass through 
, 
, 
and are parallelly with 
, 
, 
respectively are concurrently (Toshio Seimiya).
Proof. This problem is a particular case of PP4 when , , are the tangent points of the incircle with the sides of and , , are the
tangent points of-exincircle , 
-exincircle, 
-exincircle with the sides , , respectively because 
, 
, 
.
PP7. Let be a triangle with the orthocenter and the circumcircle 
. Denote 
,
and 
. Prove that 
.
Proof 1 (metric). Denote the diameter![$[AA']$](//latex.artofproblemsolving.com/f/8/3/f83bdc8e170d3bc867097c603fa03cf6edefbb4b.png)
in . Since 
is a parallelogram obtain that is the midpoint of 
. In the cyclical quadrilateral 
exists
relation
. Since 
is an harmonical division obtain 
. From the relations , obtain 
.
Proof 2 (synthetic). Denote the diameter in . Since is a parallelogram obtain that the point is the midpoint of .
Denote the second intersection
of with . Observe that 
. Since 
, 
and is an harmonical division obtain 
.
Proposed problem (extension). Let be a triangle. For a point 
denote the second intersection of with ,
the point
for which 
and the intersections 
, 
, 
.
Consider the point
so that 
and the second intersection 
of 
with . Prove that .
Proof. Denote the second intersection of 
with . Observe that 
. Since ,
and is an harmonical division obtain .
Proposed problem. Let
be a trapezoid, where 
and 
. For a point 
denote 
for which
, 
and for which 
. Prove that 
.
Proof. Denote
. Thus, 
. The division 
is harmonically,
i.e.
. Therefore, 
.
to the circle 
in the point 
. Let 
be a triangle with the circumcircle . For an interior point 
w.r.t. 
denotethe second intersections
, 
, 
of 
, 
, 
with the circle and 
, 
, 
. Prove that 
. Proof. Denote
. From the well-known relations 
and 
obtain
. Show analogously 
and 
. Apply the Menelaus' theorem :
because 
. In conclusion, .PP2 (Матматика в школе, nr.4/1984, pr. 2760). Let
with circumcirle 
. Denote 
. Prove that .Proof. Denote the orthocenter
of and 
. Observe that the circumcircle of 
is the circle with diameter ![$[OH]$](http://latex.artofproblemsolving.com/7/b/8/7b812c672802ddc509741f60b6fd241be97668af.png)
and
.Therefore, the points 
, 
, 
are inversely to the points 
, 
, 
respectively in the inversionwith the modulus
and the pole 
which belongs to the circumcircle of . In conclusion, and 
. PP3. Let
be a hexagon which is inscribed in a circle . For a point 
define the points 
,
, 
. Prove that 
.Proof. Apply the Pascal's theorem to the following cyclical hexagons :
.PP4. Consider
, 
so that 
and 
. For 
define 
, 
,
, 
. Prove that 
, 
, 
are concurrently in for which 
and 
(V.N.).Proof. Denote
, 
. Apply the Menelaus' theorem to the transversals :
.
.
. From the relation 
obtain 
.Thus,
. From 
obtain 
and from 
obtain 
, i.e. 
. In conclusion, 
.Remark. Can prove that the points
coincides with only with the relation which is independently by the relation . Indeed, apply the Ceva's' theorem to :
. From the definition of the point obtain
. Therefore, 
, i.e. 
. The position of the fixed point 
is independently by the choosing of the points
, for which 
and the slope of the line 
, i.e. this problem has three degrees of freedom. PP5. Let
be a triangle. Consider the pairs of isotomic points ![$\{D,D'\}\subset [BC]$](http://latex.artofproblemsolving.com/9/6/8/968e5a9b8adc151035f8b67c7b1ceb7a8b72ca8d.png)
, ![$\{E,E'\}\subset [CA]$](http://latex.artofproblemsolving.com/7/f/5/7f57713459b458e4e59d6945e2cb05d3031898cd.png)
, ![$\{F,F'\}\subset [AB]$](http://latex.artofproblemsolving.com/a/b/b/abb989c121c9b1d08c08621abe9016a64400fbf3.png)
. Prove that
the parallels through the points 
, 
, 
to the lines 
, 
, 
respectively are concurrently (V.N.).Proof.
PP6. Let
be a triangle with the incircle 
which touches its sides in 
, 
, 
. Denote the midpoints , , of the sides![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, ![$[CA]$](http://latex.artofproblemsolving.com/4/5/c/45c1acd47628de406680d04c09fe6314c3847acf.png)
, ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
respectively. Prove that the lines which pass through 
, 
, 
and are parallelly with 
, 
, 
respectively are concurrently (Toshio Seimiya).Proof. This problem is a particular case of PP4 when
, , are the tangent points of the incircle with the sides of and , , are thetangent points of
-exincircle , 
-exincircle, 
-exincircle with the sides , , respectively because 
, 
, 
. PP7. Let
be a triangle with the orthocenter and the circumcircle 
. Denote 
,
and 
. Prove that 
.Proof 1 (metric). Denote the diameter
![$[AA']$](http://latex.artofproblemsolving.com/f/8/3/f83bdc8e170d3bc867097c603fa03cf6edefbb4b.png)
in . Since 
is a parallelogram obtain that is the midpoint of 
. In the cyclical quadrilateral 
existsrelation
. Since 
is an harmonical division obtain 
. From the relations , obtain 
. Proof 2 (synthetic). Denote the diameter
in . Since is a parallelogram obtain that the point is the midpoint of .Denote the second intersection
of with . Observe that 
. Since 
, 
and
is an harmonical division obtain 
. Proposed problem (extension). Let
be a triangle. For a point 
denote the second intersection of with ,the point
for which 
and the intersections 
, 
, 
.Consider the point
so that 
and the second intersection 
of 
with . Prove that . Proof. Denote the second intersection
of 
with . Observe that 
. Since , and
is an harmonical division obtain . Proposed problem. Let
be a trapezoid, where 
and 
. For a point 
denote 
for which
, 
and for which 
. Prove that 
.Proof. Denote
. Thus, 
. The division 
is harmonically,i.e.
. Therefore, 
. This post has been edited 90 times. Last edited by Virgil Nicula, Dec 1, 2015, 10:43 AM
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