415. Geometry 8.
by Virgil Nicula, Mar 21, 2015, 2:09 PM
Lemma (Sunken Rock). Let a circle 
and 
. Denote 
where 
is the tangent line to at 
. Let 
be a point
for what
doesn't separate 
and 
. Denote the projections 
of on ![$([AB],[BT],[AT])$](//latex.artofproblemsolving.com/f/0/c/f0c4cd45e0e425712174145e2952dc77a4e14e78.png)
respectively. Prove that 
.
Proof.
.
PP11. Let a cyclical
with the circumcircle and a point 
. Define the projection 
of 
on 
. Prove that 
.
Proof 1. Suppose w.l.o.g. that
separates and 
. Thus, 
. Prove analogously that 
. Indeed,
.
Proof 2. Denote the distancies
of the point to the tangent lines 
respectively where is the tangent line to at .
Apply the previous lemma
.
PP12. Let
with the incircle 
and the circles 
where
. Prove that there is a circle 
so that 
.
Proof. Let
. Thus, 
where 
. Observe that 
and
where 
, i.e. there is a circle 
so that .
PP13 (Elberling Vargas Diaz). Let a semicircle
on the diameter ![$[AD]$](//latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
. Define - the tangent line to at .
For
denote 
. Prove that 
.
Proof 1 (Pedro Solis). Denote
. Therefore, 
and 
.
Remark. Let
so that 
. Thus, 
, i.e. 
is tangent to 
.
Proof 2.
and 
.
PP14. Let
be a triangle. Consider a circle 
which is tangent to the sides 
and 
of 
and which is internal tangent to the
circumcircle
of at a point 
. Let 
be the incenter of . Prove that the line 
is the bisector of the angle 
, i.e. 
.
Proof. Denote the second intersections
, 
of with 
, 
respectively. From here results 
. Apply the Pascal's theorem
to
, i.e. 
. Thus, 
the quadrilaterals
and 
are cyclically 
.
PP15. Let with the circumcircle 
and the incircle 
. Prove that the Euler's relation 
.
Proof 1.
and 
.
Proof 2 (C.D.P. - Gh. Titeica, 1935). Let the diameter![$ [NS]$](//latex.artofproblemsolving.com/e/6/8/e6810e1eb8ac4a880ee0062bc51774033cdc97e7.png)
(north-south) of the circumcircle so that 
and separates the points 
and . The incircle touches
the side![$ [CA]$](//latex.artofproblemsolving.com/b/8/3/b8309224981a62a2c0df087562b92830bd7ad7b0.png)
in the point 
. Is well-known that 
and the power of the incenter w.r.t. the circumcircle is 
, i.e. 
.
Prove easily that
, i.e. 
This proof is the shortest and the oldest!
PP16. The incircle of the -isosceles touches it at 
.
Let
and 
. Suppose that 
. Ascertain the ratio 
.
Lemma. The incircle of touches it at . Let: 
. Then 
belong to the circle with the diameter ![$[BC]\ ;\ S\in ID$](//latex.artofproblemsolving.com/d/3/c/d3cdbbede27e6522f4695af87c8cd485b6755509.png)
(the incenter 
is the orthocenter of 
.
Proof. Let
. Observe that 
. Therefore, 
![$[NAC]=[NAB]$](//latex.artofproblemsolving.com/4/5/7/457e3c5b56fd25c6265dad6bbe7cc2b8a96c57cc.png)
, i.e. 
, 
, 
are concurrently. Prove easily that the quadrilaterals 
and 
are cyclically because 
. The point 
is the radical center of the circles with the diameters ![$[IB]$](//latex.artofproblemsolving.com/4/f/b/4fbcdccbee7a86b416a11fc4628f3283e3ce4976.png)
, ![$[IC]$](//latex.artofproblemsolving.com/7/6/6/7667108e1ea2f2a718a614bc59a399d56023e588.png)
, ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, i.e. the point is the orthocenter of 
.
Remark. Using this lemma (the point is even the middle of the side ![$[AC]$](//latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
), the proposed problem becomes the following very easy problem ...
In the- isosceles let 
. Prove that 
.
Proof. Denote
, 
, 
and 
, i.e. 
. Thus, 
, i.e. 
.
.
Therefore, 
and in the both cases 
. In conclusion, for the initial proposed problem 
.
PP17. Let a scalen with the circumcircle . The incircle is tangent to 
, 
, at 
, 
, 
respectively and 
- centroid of 
. Prove that 
.
Proof 1 (metric). Suppose w.l.o.g.
. Define the midpoint of and 
. The following relations are well-known or can prove easily :
.
.
Apply the Menelaus' theorem to the transversal 
and 
: 
.
![$\frac{(b-c)(p-a)}{(b+c)[2R(p-a)-r(b+c)]}$](//latex.artofproblemsolving.com/d/3/f/d3f15fe0e7f7ee1e971a937da55070d782367561.png)
and
![$\boxed{SM=(b-c)\cdot \frac{2R(p-a)-ar}{2[2R(p-a)-r(b+c)]}}\ \ (3)$](//latex.artofproblemsolving.com/3/0/7/3079b2e9e32bac1531b4fbe2e124f7059e2553eb.png)
.
Therefore, 
.
Lemma. Let with the orthocenter 
and the circumcircle 
. Then there is the Sylvester's identity 
.
Remark. If we"ll choose
as the origin of the vectorial system, i.e. 
and 
, then Sylvester's identity becomes 
.
Proof 2 (Gemath). Denote the second intersections
of the lines 
with the circumcircle of . Prove easily that the incenter of
is the orthocenter of
. From 
and 
obtain 
a.s.o.
Thus,
, what means . If 
is the orthocenter of ,
then
(the line 
is the Euler's line for ) and the previous relation becomes 
.
PP18. Let be an interior point of for which 
and 
Denote 
the symmetrical point of w.r.t. 
the circumcircles
and 
of 
and 
respectively 
the intersections 
and 
Prove that
is the midpoint of ![$[BC]\ ;\ BDCE$](//latex.artofproblemsolving.com/b/7/c/b7c9505a187b01bf21b4da7b4d0fd1b04444819b.png)
is a cyclical quadrilateral 
is the -symmedian in the triangle 
Proof.
, i.e. is the midpoint of . Remark that is a common tangent of 
and 
.
is a parallelogram, i.e. 
and 
. Remark that 
.
.
. Prove easily that 
. Thus, 
, i.e. 
is the -symmedian in the triangle
PP19 (JBMO 2015). Let an acute
and 
. Let 
. Prove that 
.
Proof.
. Thus, the quadrilaterals 
and 
are cyclically. Therefore, 
In conclusion, . Nice problem !
PP20. Let which isn't isosceles. Construct the triangles 
, 
such that 
and 
. Prove that 
.
Proof 1 . Let
. Therefore, 
. Suppose w.l.o.g. 
. Observe that
.
Proof 2 (Sunken Rock). a solution avoiding boring calculations: take so that 
is an isosceles trapezoid and construct the circle tangent at 
to and passing through 
; it will
intersect
at 
respectively. Since 
, it follows that the parallel through to 
intersects 
at , and 
. As 
implies
that is tangent to circle 
, thus 
or 
. Also 
or 
(as constructed, 
). From
and 
with 
we get 
, hence there is a spiral similarity centered at , sending to and to , thus being done.
PP21 (B.Komal). Prove that if
are the angles of an acute-angled 
then 
Proof. Denote
where 
and 
Thus,![$xyz=x+y+z\ge 3\sqrt[3]{xyz}\implies$](//latex.artofproblemsolving.com/8/1/4/81481154976aa36ba923cdc60c9d8a792caeeab7.png)
Hence 
PP22 (Ruben Dario). Let with incircle 
and -excircle 
Denote 
and 
Prove that 
Proof. Is well known that
and 
a.s.o. Therefore, 
Apply
a remarkable identity
In conclusion, from the relations and obtain the required identity.
and 
. Denote 
where 
is the tangent line to at 
. Let 
be a pointfor what
doesn't separate 
and 
. Denote the projections 
of on ![$([AB],[BT],[AT])$](http://latex.artofproblemsolving.com/f/0/c/f0c4cd45e0e425712174145e2952dc77a4e14e78.png)
respectively. Prove that 
.Proof.
.PP11. Let a cyclical
with the circumcircle and a point 
. Define the projection 
of 
on 
. Prove that 
.Proof 1. Suppose w.l.o.g. that
separates and 
. Thus, 
. Prove analogously that 
. Indeed,
.Proof 2. Denote the distancies
of the point to the tangent lines 
respectively where is the tangent line to at .Apply the previous lemma
. PP12. Let
with the incircle 
and the circles 
where
. Prove that there is a circle 
so that 
.Proof. Let
. Thus, 
where 
. Observe that 
and
where 
, i.e. there is a circle 
so that .PP13 (Elberling Vargas Diaz). Let a semicircle
on the diameter ![$[AD]$](http://latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
. Define - the tangent line to at .For
denote 
. Prove that 
.Proof 1 (Pedro Solis). Denote
. Therefore, 
and 
.Remark. Let
so that 
. Thus, 
, i.e. 
is tangent to 
.Proof 2.
and 
.PP14. Let
be a triangle. Consider a circle 
which is tangent to the sides 
and 
of 
and which is internal tangent to thecircumcircle
of at a point 
. Let 
be the incenter of . Prove that the line 
is the bisector of the angle 
, i.e. 
.Proof. Denote the second intersections
, 
of with 
, 
respectively. From here results 
. Apply the Pascal's theoremto
, i.e. 
. Thus, 
the quadrilaterals
and 
are cyclically 
.PP15. Let
with the circumcircle 
and the incircle 
. Prove that the Euler's relation 
.Proof 1.
and 
.
Proof 2 (C.D.P. - Gh. Titeica, 1935). Let the diameter
![$ [NS]$](http://latex.artofproblemsolving.com/e/6/8/e6810e1eb8ac4a880ee0062bc51774033cdc97e7.png)
(north-south) of the circumcircle so that 
and separates the points 
and . The incircle touchesthe side
![$ [CA]$](http://latex.artofproblemsolving.com/b/8/3/b8309224981a62a2c0df087562b92830bd7ad7b0.png)
in the point 
. Is well-known that 
and the power of the incenter w.r.t. the circumcircle is 
, i.e. 
.Prove easily that
, i.e. 
This proof is the shortest and the oldest!PP16. The incircle
of the -isosceles touches it at 
.Let
and 
. Suppose that 
. Ascertain the ratio 
.Lemma. The incircle
of touches it at . Let: 
. Then 
belong to the circle with the diameter ![$[BC]\ ;\ S\in ID$](http://latex.artofproblemsolving.com/d/3/c/d3cdbbede27e6522f4695af87c8cd485b6755509.png)
(the incenter 
is the orthocenter of 
.Proof. Let
. Observe that 
. Therefore, 
![$[NAC]=[NAB]$](http://latex.artofproblemsolving.com/4/5/7/457e3c5b56fd25c6265dad6bbe7cc2b8a96c57cc.png)
, i.e. 
, 
, 
are concurrently. Prove easily that the quadrilaterals 
and 
are cyclically because 
. The point 
is the radical center of the circles with the diameters ![$[IB]$](http://latex.artofproblemsolving.com/4/f/b/4fbcdccbee7a86b416a11fc4628f3283e3ce4976.png)
, ![$[IC]$](http://latex.artofproblemsolving.com/7/6/6/7667108e1ea2f2a718a614bc59a399d56023e588.png)
, ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, i.e. the point is the orthocenter of 
.Remark. Using this lemma (the point
is even the middle of the side ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
), the proposed problem becomes the following very easy problem ... In the
- isosceles let 
. Prove that 
.Proof. Denote
, 
, 
and 
, i.e. 
. Thus, 
, i.e. 
.
.Therefore,
and in the both cases 
. In conclusion, for the initial proposed problem 
.PP17. Let a scalen
with the circumcircle . The incircle is tangent to 
, 
, at 
, 
, 
respectively and 
- centroid of 
. Prove that 
.Proof 1 (metric). Suppose w.l.o.g.
. Define the midpoint of and 
. The following relations are well-known or can prove easily :
.
.
Apply the Menelaus' theorem to the transversal 
and 
: 
.
![$\frac{(b-c)(p-a)}{(b+c)[2R(p-a)-r(b+c)]}$](http://latex.artofproblemsolving.com/d/3/f/d3f15fe0e7f7ee1e971a937da55070d782367561.png)
and
![$\boxed{SM=(b-c)\cdot \frac{2R(p-a)-ar}{2[2R(p-a)-r(b+c)]}}\ \ (3)$](http://latex.artofproblemsolving.com/3/0/7/3079b2e9e32bac1531b4fbe2e124f7059e2553eb.png)
.
Therefore, 
.Lemma. Let
with the orthocenter 
and the circumcircle 
. Then there is the Sylvester's identity 
.Remark. If we"ll choose
as the origin of the vectorial system, i.e. 
and 
, then Sylvester's identity becomes 
.Proof 2 (Gemath). Denote the second intersections
of the lines 
with the circumcircle of . Prove easily that the incenter of is the orthocenter of
. From 
and 
obtain 
a.s.o.Thus,
, what means . If 
is the orthocenter of ,then
(the line 
is the Euler's line for ) and the previous relation becomes 
. PP18. Let
be an interior point of for which 
and 
Denote 
the symmetrical point of w.r.t. 
the circumcircles
and 
of 
and 
respectively 
the intersections 
and 
Prove that
is the midpoint of ![$[BC]\ ;\ BDCE$](http://latex.artofproblemsolving.com/b/7/c/b7c9505a187b01bf21b4da7b4d0fd1b04444819b.png)
is a cyclical quadrilateral 
is the -symmedian in the triangle 
Proof.
, i.e. is the midpoint of . Remark that is a common tangent of 
and 
.
is a parallelogram, i.e. 
and 
. Remark that 
.
. . Prove easily that 
. Thus, 
, i.e. 
is the -symmedian in the triangle PP19 (JBMO 2015). Let an acute
and 
. Let 
. Prove that 
.Proof.
. Thus, the quadrilaterals 
and 
are cyclically. Therefore, 
In conclusion, . Nice problem !PP20. Let
which isn't isosceles. Construct the triangles 
, 
such that 
and 
. Prove that 
.Proof 1 . Let
. Therefore, 
. Suppose w.l.o.g. 
. Observe that
.Proof 2 (Sunken Rock). a solution avoiding boring calculations: take
so that 
is an isosceles trapezoid and construct the circle tangent at 
to and passing through 
; it willintersect
at 
respectively. Since 
, it follows that the parallel through to 
intersects 
at , and 
. As 
impliesthat
is tangent to circle 
, thus 
or 
. Also 
or 
(as constructed, 
). From
and 
with 
we get 
, hence there is a spiral similarity centered at , sending to and to , thus being done.PP21 (B.Komal). Prove that if
are the angles of an acute-angled 
then 
Proof. Denote
where 
and 
Thus,
![$xyz=x+y+z\ge 3\sqrt[3]{xyz}\implies$](http://latex.artofproblemsolving.com/8/1/4/81481154976aa36ba923cdc60c9d8a792caeeab7.png)
Hence 
PP22 (Ruben Dario). Let
with incircle 
and -excircle 
Denote 
and 
Prove that 
Proof. Is well known that
and 
a.s.o. Therefore, 
Applya remarkable identity
In conclusion, from the relations and obtain the required identity.This post has been edited 108 times. Last edited by Virgil Nicula, Nov 11, 2016, 9:19 AM
August 2018
January 2018