462. Diverse probleme de geometrie.
by Virgil Nicula, Aug 3, 2018, 8:20 AM
P1. Let 
with the circumcircle 
and with the incircle 
. Define the points 
. Prove that 
.
Proof. Denote the power
of 
w.r.t. the circle 
. Thus, 
.
The relations
a.s.o. are well-known. Apply the Stewart's theorem to the rays 
in 
respectively :
. Hence 
, i.e. .
P2 (IMO 2010). Let an interior
of . Lines 
meet again the circumcircle 
at 
respectively. The tangent at 
to meets 
at 
. Show that 
.
Proof. Consider the circle which is tangent to in and to 
in . Denote 
, where 
. From the well-known property obtain 
,
i.e.
. Since 
obtain 
(the line is tangent to the circumcircle of 
) , i.e. 
.
In conclusion,
, 
, i.e. 
. Nice problem !
P3 (Balcan geo 2003). Let and the tangent to the circumcircle of at 
meet 
at 
. The perpendicular to at 
meets the perpendicular
bisector of
at 
. The perpendicular to at 
meets the perpendicular bisector of 
at 
. Prove that the points 
. (Valentin Vornicu).
Proof (Virgil NICULA).
.
P4 (OME, 2009). Let
be an acute triangle with the incircle 
and the circumcircle 
.
Denote
for which 
and 
. Prove that 
.
Proof. From the identity
for 
obtain 
.
Therefore,
. In conclusion, 
, i.e. 
.
Extension (Virgil NICULA). Let an acute with the incircle 
and the circumcircle . Let
so that , 
and 
, 
. Prove that 
.
Particular cases.
what are evidently. Thus we
verified the upper identity ! Also we can find easily the points
for which 
, i.e. 
.
Proof. Let
for which 
. Apply the generalized Pythagoras' in 
: 
![$ \left[sx - (s - a)h_a\right]\left(x - h_a\right) + h_a(s - a)\left(2R - h_a\right) =$](//latex.artofproblemsolving.com/7/9/e/79eda64de53c040d2adf3e07dc1a231e7f8e83ea.png)
. Since 
obtain 
.
P5 (USA TST 2006). Let be a triangle. Triangles 
and 
are constructed outside of triangle such that 
and 
. Segments 
and 
meet at 
. Let 
be the circumcenter of triangle 
. Prove that 
Proof. I"ll show that
, i.e. 
. Let 
, the circumcircle 
of 
and 
. Thus, 
the quadrilaterals 
, 
are cyclically with the radical axis 
. Observe that 
the
ray
is the bisector of the angle 
. Prove easily that 
. Thus, 
and 
. Apply the generalized Pythagoras'
to
, 
.
I used the following simple identities
and 
.
P6. For and 
construct the parallelogram 
. Let 
.
Prove that the circumcenter of
belongs to the circumcircle of 
.
Proof.
P7 (France TST 2007). Let such that 
. Let ![$D\in [AC]$](//latex.artofproblemsolving.com/c/5/a/c5acd77baad8d15d98170c2ac60f572eeeed5507.png)
such that 
. The incircle 
of
touches its sides in![$K\in [AB]$](//latex.artofproblemsolving.com/f/7/c/f7c55ea4f0642b6ee4395cbe8c000c6fec53e3ef.png)
and ![$L\in [AC]$](//latex.artofproblemsolving.com/c/4/5/c45e50a5a8063a6607d6afd834b15cea829eb243.png)
. Let 
be the incircle of 
. Denote 
. Prove that 
.
Proof.. The incircle
touches in 
, 
and 
. Observe that 
. Thus,
. Since 
obtain that 
. Denote 
. The property 
is well-known (see the lower remark). Thus,
is cyclically. Analogously 
, i.e. 
. Observe that 
. Apply the Menelaus' theorem to transversal 
in 
: 
.
So 
the quad. 
is cyclically 
or 
P8. Let
be a triangle with 
. Let 
bisect 
and let 
bisect 
, where 
and 
. If 
, then what are the angles of the triangle ?
Proof I (metric). Let
. Thus, 
. Thus, 
, 
because .
Proof II (trigonometric) Denote
. I"ll apply the Sinus' theorem in the triangles 
, 
: 
![$ \sin (60+x)\left[\sin (30+2x)+\frac 12\right]=\sin (30+2x)(\sin x+\sin 60)$](//latex.artofproblemsolving.com/f/9/0/f9014e97ba223e408eef58d3d00ae7e6856a69a3.png)
![$ [\cos (x-30)-\cos (90-2x)]+$](//latex.artofproblemsolving.com/6/c/f/6cfc307118123ab15122148835ecb7deb5cf7715.png)
![$[\sin 3x+\sin (60-3x)]=$](//latex.artofproblemsolving.com/9/4/f/94fe11441f683123ef54d217847bfd74ccc07ca4.png)
![$ [\cos (x+30)-\cos (x-30)]+\cos (2x-30)$](//latex.artofproblemsolving.com/b/0/5/b05b68ab2f09bc2aa220f250038619fa84e1d0e7.png)
![$2\sin\left(60-\frac {3x}{2}\right)\left[\sin\left(30+\frac x2\right)-\sin\left(30-\frac x2\right)\right]$](//latex.artofproblemsolving.com/d/6/7/d6787a30786e7e73494cf66e267856c792fca8d8.png)
.
Thus,
or 
.
Extension (Virgil NICULA). Let an acute with the incircle and the circumcircle . Let
so that , and , . Prove that .
with the circumcircle 
and with the incircle 
. Define the points 
. Prove that 
.Proof. Denote the power
of 
w.r.t. the circle 
. Thus, 
.The relations
a.s.o. are well-known. Apply the Stewart's theorem to the rays 
in 
respectively :
. Hence 
, i.e. .P2 (IMO 2010). Let an interior
of . Lines 
meet again the circumcircle 
at 
respectively. The tangent at 
to meets 
at 
. Show that 
.Proof. Consider the circle which is tangent to
in and to 
in . Denote 
, where 
. From the well-known property obtain 
,i.e.
. Since 
obtain 
(the line is tangent to the circumcircle of 
) , i.e. 
.In conclusion,
, 
, i.e. 
. Nice problem !P3 (Balcan geo 2003). Let
and the tangent to the circumcircle of at 
meet 
at 
. The perpendicular to at 
meets the perpendicularbisector of
at 
. The perpendicular to at 
meets the perpendicular bisector of 
at 
. Prove that the points 
. (Valentin Vornicu).Proof (Virgil NICULA).
.P4 (OME, 2009). Let
be an acute triangle with the incircle 
and the circumcircle 
.Denote
for which 
and 
. Prove that 
.Proof. From the identity
for 
obtain 
.Therefore,
. In conclusion, 
, i.e. 
.Extension (Virgil NICULA). Let an acute
with the incircle 
and the circumcircle . Let so that
, 
and 
, 
. Prove that 
.Particular cases.
what are evidently. Thus weverified the upper identity ! Also we can find easily the points
for which 
, i.e. 
.Proof. Let
for which 
. Apply the generalized Pythagoras' in 
: 
![$ \left[sx - (s - a)h_a\right]\left(x - h_a\right) + h_a(s - a)\left(2R - h_a\right) =$](http://latex.artofproblemsolving.com/7/9/e/79eda64de53c040d2adf3e07dc1a231e7f8e83ea.png)
. Since 
obtain 
.P5 (USA TST 2006). Let
be a triangle. Triangles 
and 
are constructed outside of triangle such that 
and 
. Segments 
and 
meet at 
. Let 
be the circumcenter of triangle 
. Prove that 
Proof. I"ll show that
, i.e. 
. Let 
, the circumcircle 
of 
and 
. Thus, 
the quadrilaterals 
, 
are cyclically with the radical axis 
. Observe that 
theray
is the bisector of the angle 
. Prove easily that 
. Thus, 
and 
. Apply the generalized Pythagoras'to
, 
.I used the following simple identities
and 
.P6. For
and 
construct the parallelogram 
. Let 
.Prove that the circumcenter of
belongs to the circumcircle of 
.Proof.
P7 (France TST 2007). Let
such that 
. Let ![$D\in [AC]$](http://latex.artofproblemsolving.com/c/5/a/c5acd77baad8d15d98170c2ac60f572eeeed5507.png)
such that 
. The incircle 
of touches its sides in
![$K\in [AB]$](http://latex.artofproblemsolving.com/f/7/c/f7c55ea4f0642b6ee4395cbe8c000c6fec53e3ef.png)
and ![$L\in [AC]$](http://latex.artofproblemsolving.com/c/4/5/c45e50a5a8063a6607d6afd834b15cea829eb243.png)
. Let 
be the incircle of 
. Denote 
. Prove that 
.Proof.. The incircle
touches in 
, 
and 
. Observe that 
. Thus,
. Since 
obtain that 
. Denote 
. The property 
is well-known (see the lower remark). Thus,
is cyclically. Analogously 
, i.e. 
. Observe that 
. Apply the Menelaus' theorem to transversal 
in 
: 
.So
the quad. 
is cyclically 
or 
P8. Let
be a triangle with 
. Let 
bisect 
and let 
bisect 
, where 
and 
. If 
, then what are the angles of the triangle ?Proof I (metric). Let
. Thus, 
. Thus, 
, 
because .Proof II (trigonometric) Denote
. I"ll apply the Sinus' theorem in the triangles 
, 
: 
![$ \sin (60+x)\left[\sin (30+2x)+\frac 12\right]=\sin (30+2x)(\sin x+\sin 60)$](http://latex.artofproblemsolving.com/f/9/0/f9014e97ba223e408eef58d3d00ae7e6856a69a3.png)
![$ [\cos (x-30)-\cos (90-2x)]+$](http://latex.artofproblemsolving.com/6/c/f/6cfc307118123ab15122148835ecb7deb5cf7715.png)
![$[\sin 3x+\sin (60-3x)]=$](http://latex.artofproblemsolving.com/9/4/f/94fe11441f683123ef54d217847bfd74ccc07ca4.png)
![$ [\cos (x+30)-\cos (x-30)]+\cos (2x-30)$](http://latex.artofproblemsolving.com/b/0/5/b05b68ab2f09bc2aa220f250038619fa84e1d0e7.png)
![$2\sin\left(60-\frac {3x}{2}\right)\left[\sin\left(30+\frac x2\right)-\sin\left(30-\frac x2\right)\right]$](http://latex.artofproblemsolving.com/d/6/7/d6787a30786e7e73494cf66e267856c792fca8d8.png)
.Thus,
or 
.Extension (Virgil NICULA). Let an acute
with the incircle and the circumcircle . Let so that
, and , . Prove that .This post has been edited 134 times. Last edited by Virgil Nicula, Mar 28, 2019, 1:43 PM
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