279. O.M. Holland , O.M. Ireland 1988 and TST USA 2000, etc.
by Virgil Nicula, May 21, 2011, 2:19 AM
PP1 (Holland). Let 
be a triangle with 
. Denote 
and the symmetrical 
of 
w.r.t. 
. Prove that 
.
Proof 1. Denote the symmetrical point
of the point w.r.t. 
and 
. Since 
is -right angled and
results that the median 
from is homologously in similitude with the median 
from
, i.e. 
is cyclic 
.
Proof 2. Denote the midpoint
of ![$[AD]$](//latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
. Observe that 
is 
-altitude in
the point is the orthocenter of 
.
Proof 3. Denote the points
and 
so that 
. Observe that 
is a parallelogram 
, i.e. is the orthocenter of 
.
Proof 4. Denote
. Observe that 
, i.e. 
. From the well-known relation 
, i.e. 
. Since 
from 
and 
obtain that 
, i.e. .
An easy extension. Let be an -right angled triangle. Denote the point 
for which 
.
Consider the points
and 
so that 
and 
. Prove that .
Proof 1. Denote . Observe that 
. From the well-known relation 
obtain that
. Since 
obtain that 
. From the relations and
obtain that
, what is truly.
Proof 2. Let
and 
for which 
and 
. So 
.
Denote
. Then 
. In 
we have 
and 
.
PP2 (easy - Holland). Let be an -isosceles triangle. Consider two points 
and 
so that
. Prove that the circumcenter of 
belongs to the circumcircle of 
.
Proof. Denote the circumcircle
of . Observe that 
, i.e. the circumcenter 
of the triangle belongs to the circumcircle of .
PP3 (Ireland - 1988). Let
be a point of the small arc 
in the circumircle of the square 
. Prove that 
.
Proof 1 (metric). Apply Ptolemy's theorem to
.
Proof 2 (trigonometric). Denote
. Observe that 
(degrees) and the required relation 
is equivalently with the trigonometrical relation
. Since 
prove easily that
and 
, i.e. 
.
An easy extension. Let be a point of the small arc in the circumircle of the isosceles trapezoid
with 
and 
. Prove that .
Proof (metric). Apply Ptolemy's theorem to
.
A similar problem. Let be an -isosceles triangle with the circumcircle 
. Consider two mobile points 
so that the sideline
separates , 
and doesn't separate , . Prove that the ratio ![$\frac {[BPCQ]}{AQ^2-AP^2}$](//latex.artofproblemsolving.com/e/4/0/e409021322e11cbef8a3f86d4f77f8bfad9f45c4.png)
is constant.
Proof. Denote
so that ![$[AA']$](//latex.artofproblemsolving.com/f/8/3/f83bdc8e170d3bc867097c603fa03cf6edefbb4b.png)
is a diameter of . Construct 
so that 
, 
. Apply the Ptolemy's theorem
to the quadrilaterals
. Observe that the relation 
. Thus, 
and![$\frac {[BPCQ]}{AQ^2-AP^2}=$](//latex.artofproblemsolving.com/8/8/d/88df5a44f4fd8f01370ea197fb0e0367d4626e3d.png)
![$\frac {[BPC]+[BQC]}{AQ^2-AP^2}=$](//latex.artofproblemsolving.com/e/c/4/ec49e99e7072454675fbd42d03ab13db7a020f96.png)
(constant).
PP4 - Turkey TST 2011. Let with incenter 
and circumcircle . Denote diameter of and 
. Show that 
.
Proof. Observe that
and 
. Apply the Pythagoras' relation in the mentioned triangles :
In conclusion,
.
PP5 - USA TST 2000. Let
be a cyclic quadrilateral and let 
and 
be the feet of perpendiculars from the intersection of diagonals
and 
to 
and 
respectively. Prove that 
is perpendicular to the line through the midpoints of 
and 
.
Lemma. Construct outside of the right triangles 
. Denote the midpoint 
of ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, Prove that 
.
Proof. Denote
and the midpoints , 
of , ![$[CD]$](//latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
respectively. Apply above lemma to 
and its exterior right 
, 
.
Obtain . Apply again above lemma to 
and its exterior right 
, 
. Obtain 
.In conclusion, 
.
Another proof. Let , be the midpoints of 
, respectively. Let be the intersection of diagonals. Let , 
be the reflections of over 
,
respectively. We know that
and 
are directly similar and are kites. Then the midpoints of the lines connecting corresponding points of the two
similar figures forms a figure similar to the original, i.e.
is similar to both and . Therefore is a kite and 
.
PP6 - Polosh & Costa Rica 2006. Let be an acute triangle. Its incircle 
touches it at the points , 
, . Denote:
the centroid of the triangle ;
the A-exincircle 
, the B-exincircle 
, the C-exincircle 
;
the Nagel's point 
; 
, 
, 
;
the points 
, 
, 
;
the reflections 
, 
, 
of the points , , with respect to the point .
Lemma 1.
.
Proof.
![$\frac 19\left\{\left[2\left(a^2+c^2\right)-b^2\right]-\left[2\left(a^2+b^2\right)-c^2\right]\right\}$](//latex.artofproblemsolving.com/3/2/9/329f57c95e21b647aae442d05502ce94fa6aeac4.png)
.
Remark.
and 
. Therefore, 
.
Lemma 2.
;
; 
and 
a.s.o.
Proof. Apply the Stewart's relation for the Nagel-ray
in the triangle and obtain the relation .
Apply the van Aubel's relation
and obtain the relation . 
, i.e. the relations .
Remark.
.
An equivalent and complete enunciation of the proposed problem. Let be an acute triangle for which 
. Then
are cyclically 
and in this case 
belong to
the circle
, where the point is the second intersection between the line 
and the circumcircle of the triangle .
Proof (indication). The points are cyclically 
a.s.o.
Example.
, 
, 
. Remark. 
.
PP7 - 32nd BMO 1996. Let
for which 
.
Proof.
![$4\sqrt[4]{s_4}\ge 12=s_1\ge 4\sqrt[4]{s_4}\implies$](//latex.artofproblemsolving.com/6/e/7/6e73ba959481dcc46e127b718877a12a90f20b96.png)
![$\frac {s_1}{4}=\sqrt[4]{s_4}\implies$](//latex.artofproblemsolving.com/d/3/a/d3a3bd98ce762ccb71efdb5e09ec192ff6fdf3f8.png)
.
PP8 - 8th China Southeast MO. In , for an interior denote 
and 
. Construct similarly the points 
and 
. Prove that 
.
Proof. Denote
, 
and 
. Suppose w.l.o.g. 
. Obtain that 
and 
. Apply Menelaus' teorem
to transversal
. Analogously 
and 
. Hence .
PP9 (Iran TST 2007, Day 2). Let
be the incircle of the triangle . Let 
and 
be two points such that 
and 
is tangent to the circle . The
lines, 
touch the circle at , respectively. Denote the middlepoint of ![$[PQ]$](//latex.artofproblemsolving.com/2/1/c/21ca08816cf8b23ddf756ce9ae098ad327f2443d.png)
and 
. Prove that the line 
is tangent to the circle (Ali Khezeli)
Proof. Denote the intersection
and the middlepoints 
, 
of the segments , respectively. Observe that 
, 
. From the well-known properties
and 
obtain immediatelly that 
, i.e. 
and the line is tangent to the circle because the lines , 
are symmetrically w.r.t the line 
.
be a triangle with 
. Denote 
and the symmetrical 
of 
w.r.t. 
. Prove that 
.Proof 1. Denote the symmetrical point
of the point w.r.t. 
and 
. Since 
is -right angled and
results that the median 
from is homologously in similitude with the median 
from
, i.e. 
is cyclic 
.Proof 2. Denote the midpoint
of ![$[AD]$](http://latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
. Observe that 
is 
-altitude in
the point is the orthocenter of 
.Proof 3. Denote the points
and 
so that 
. Observe that 
is a parallelogram 
, i.e. is the orthocenter of 
.Proof 4. Denote
. Observe that 
, i.e. 
. From the well-known relation 
, i.e. 
. Since 
from 
and 
obtain that 
, i.e. .An easy extension. Let
be an -right angled triangle. Denote the point 
for which 
.Consider the points
and 
so that 
and 
. Prove that .Proof 1. Denote
. Observe that 
. From the well-known relation 
obtain that
. Since 
obtain that 
. From the relations and obtain that
, what is truly.Proof 2. Let
and 
for which 
and 
. So 
.Denote
. Then 
. In 
we have 
and 
.PP2 (easy - Holland). Let
be an -isosceles triangle. Consider two points 
and 
so that
. Prove that the circumcenter of 
belongs to the circumcircle of 
.Proof. Denote the circumcircle
of . Observe that 
, i.e. the circumcenter 
of the triangle belongs to the circumcircle of .PP3 (Ireland - 1988). Let
be a point of the small arc 
in the circumircle of the square 
. Prove that 
.Proof 1 (metric). Apply Ptolemy's theorem to
.Proof 2 (trigonometric). Denote
. Observe that 
(degrees) and the required relation 
is equivalently with the trigonometrical relation
. Since 
prove easily that
and 
, i.e. 
.An easy extension. Let
be a point of the small arc in the circumircle of the isosceles trapezoid with 
and 
. Prove that .Proof (metric). Apply Ptolemy's theorem to
.A similar problem. Let
be an -isosceles triangle with the circumcircle 
. Consider two mobile points 
so that the sideline
separates , 
and doesn't separate , . Prove that the ratio ![$\frac {[BPCQ]}{AQ^2-AP^2}$](http://latex.artofproblemsolving.com/e/4/0/e409021322e11cbef8a3f86d4f77f8bfad9f45c4.png)
is constant.Proof. Denote
so that ![$[AA']$](http://latex.artofproblemsolving.com/f/8/3/f83bdc8e170d3bc867097c603fa03cf6edefbb4b.png)
is a diameter of . Construct 
so that 
, 
. Apply the Ptolemy's theoremto the quadrilaterals
. Observe that the relation 
. Thus, 
and
![$\frac {[BPCQ]}{AQ^2-AP^2}=$](http://latex.artofproblemsolving.com/8/8/d/88df5a44f4fd8f01370ea197fb0e0367d4626e3d.png)
![$\frac {[BPC]+[BQC]}{AQ^2-AP^2}=$](http://latex.artofproblemsolving.com/e/c/4/ec49e99e7072454675fbd42d03ab13db7a020f96.png)
(constant).PP4 - Turkey TST 2011. Let
with incenter 
and circumcircle . Denote diameter of and 
. Show that 
.Proof. Observe that
and 
. Apply the Pythagoras' relation in the mentioned triangles :
In conclusion,
.PP5 - USA TST 2000. Let
be a cyclic quadrilateral and let 
and 
be the feet of perpendiculars from the intersection of diagonals
and 
to 
and 
respectively. Prove that 
is perpendicular to the line through the midpoints of 
and 
.Lemma. Construct outside of
the right triangles 
. Denote the midpoint 
of ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, Prove that 
.
of the sides ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
, ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
respectively. Since 
.
. Observe that 
, 
and 
. Apply the generalized Pytagoras' theorem in :
.
![$2\left(b^2-c^2\right)\cdot\cos\phi =\left[\left(a^2+b^2-c^2\right)-\left(a^2+c^2-b^2\right)\right]\cdot\cos\phi$](http://latex.artofproblemsolving.com/f/d/2/fd2dc00b4463529c862b9f91654289cc59cf47b2.png)
, what is truly. Proof. Denote
and the midpoints , 
of , ![$[CD]$](http://latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
respectively. Apply above lemma to 
and its exterior right 
, 
.Obtain
. Apply again above lemma to 
and its exterior right 
, 
. Obtain 
.In conclusion, 
.Another proof. Let
, be the midpoints of 
, respectively. Let be the intersection of diagonals. Let , 
be the reflections of over 
, respectively. We know that
and 
are directly similar and are kites. Then the midpoints of the lines connecting corresponding points of the twosimilar figures forms a figure similar to the original, i.e.
is similar to both and . Therefore is a kite and 
.PP6 - Polosh & Costa Rica 2006. Let
be an acute triangle. Its incircle 
touches it at the points , 
, . Denote:
the centroid of the triangle ; the A-exincircle 
, the B-exincircle 
, the C-exincircle 
; the Nagel's point 
; 
, 
, 
; the points 
, 
, 
; the reflections 
, 
, 
of the points , , with respect to the point .Lemma 1.
.Proof.
![$\frac 19\left\{\left[2\left(a^2+c^2\right)-b^2\right]-\left[2\left(a^2+b^2\right)-c^2\right]\right\}$](http://latex.artofproblemsolving.com/3/2/9/329f57c95e21b647aae442d05502ce94fa6aeac4.png)
.Remark.
and 
. Therefore, 
.Lemma 2.
;
; 
and 
a.s.o.Proof. Apply the Stewart's relation for the Nagel-ray
in the triangle and obtain the relation .Apply the van Aubel's relation
and obtain the relation . 
, i.e. the relations .Remark.
.An equivalent and complete enunciation of the proposed problem. Let
be an acute triangle for which 
. Then
are cyclically 
and in this case 
belong tothe circle
, where the point is the second intersection between the line 
and the circumcircle of the triangle .Proof (indication). The points
are cyclically 
a.s.o.Example.
, 
, 
. Remark. 
.PP7 - 32nd BMO 1996. Let
for which 
.Proof.
![$4\sqrt[4]{s_4}\ge 12=s_1\ge 4\sqrt[4]{s_4}\implies$](http://latex.artofproblemsolving.com/6/e/7/6e73ba959481dcc46e127b718877a12a90f20b96.png)
![$\frac {s_1}{4}=\sqrt[4]{s_4}\implies$](http://latex.artofproblemsolving.com/d/3/a/d3a3bd98ce762ccb71efdb5e09ec192ff6fdf3f8.png)
. PP8 - 8th China Southeast MO. In
, for an interior denote 
and 
. Construct similarly the points 
and 
. Prove that 
.Proof. Denote
, 
and 
. Suppose w.l.o.g. 
. Obtain that 
and 
. Apply Menelaus' teoremto transversal
. Analogously 
and 
. Hence .PP9 (Iran TST 2007, Day 2). Let
be the incircle of the triangle . Let 
and 
be two points such that 
and 
is tangent to the circle . Thelines
, 
touch the circle at , respectively. Denote the middlepoint of ![$[PQ]$](http://latex.artofproblemsolving.com/2/1/c/21ca08816cf8b23ddf756ce9ae098ad327f2443d.png)
and 
. Prove that the line 
is tangent to the circle (Ali Khezeli)Proof. Denote the intersection
and the middlepoints 
, 
of the segments , respectively. Observe that 
, 
. From the well-known properties
and 
obtain immediatelly that 
, i.e. 
and the line is tangent to the circle because the lines , 
are symmetrically w.r.t the line 
.
.
,
,
.
and the triangle 
,
is the polar 
of the point 
of the point 
w.r.t. the circle 
.This post has been edited 110 times. Last edited by Virgil Nicula, Nov 22, 2015, 6:55 AM
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