403. Some nice geometry problems for the high school. II
by Virgil Nicula, Nov 10, 2014, 6:31 AM
PP15. Let 
and 
be two circles so that 
The line 
is a common tangent for 
and 
in 
and 
.
Let
be the semicircle with 
so that the circles 
are interior tangent to 
Prove that 
Proof. Apply the Pythagoras's theorem to
Therefore,
, where 
Thus, 
Let 
and 
so that 
and 
Observe that 
Apply the Pythagoras's
theorem to
From the relations 
and the relation
obtain that 
From the relations 
and 
obtain that 
. Drawing of PP15 & PP16.
PP16. Let and be two circles so that The line is a common tangent for and in and .
Let be the semicircle with and what is exterior tangent to the circles 
Prove that 
Proof. Apply the Pythagoras's theorem to
Therefore, 
Let 
and 
so that and Observe that 
Apply the Pythagoras's
theorem to
From the relations
and the relation obtain that 
From the relations and obtain that 
PP17. Let incircle
of 
and 
, 
so that 
is tangent to 
. Prove that there is the relation 
Proof. Denote
, 
and 
. Thus, ![$[AMN]=2\cdot [AIM]+2\cdot [AIN]-[AEIF]$](//latex.artofproblemsolving.com/2/0/b/20b2b442ad84abc6a9cd4f54c0881b31e842be7c.png)
, i.e. 
because
and 
. Hence 
, i.e. 
. Thus, 
.
The required relation is equivalently with
, i.e. the relation 
.
PP18. Let the incircle of , , and for ![$P\in \overarc[]{EF}$](//latex.artofproblemsolving.com/5/2/a/52aad01ba7b62fda7f1615ada91abf33dfb7627a.png)
let the lengths 
of the distancies from
to 
, 
, respectively. Prove that 
.
Proof. Denote
so that 
and 
so that 
. Thus, 
. Obtain analogously 
. Prove easily that 
. Apply the Ptolemy theorem to the cyclical quadrilateral
.
PP19. Let and 
so that 
. Prove that 
Particularly 
Proof 1. Apply Stewart's relation to cevian
in 
Using an well-known
property in
obtain that 
If 
then from obtain that
![$a\cdot MC^2+\left[\left(c^2+ac-b^2\right)-a^2\right]\cdot MC-a\left(c^2+ac-b^2\right)=0$](//latex.artofproblemsolving.com/1/4/1/141080b68876611a38498bbe09fb2bb4d2203d8d.png)
![$\left(MC-a\right)\left[aMC+c(a+c)-b^2\right]=0\iff$](//latex.artofproblemsolving.com/9/8/3/9838ae54547e2985ca3e2e4ab314d8416a2b921a.png)
PP20. Let with the orthocenter 
and the circumcircle 
. Denote
and 
. Prove that 
.
Proof. Suppose w.l.o.g. that
. Let 
so that 
is a diameter of 
and 
so that 
. Is well-known
that
is a parallelogram. Thus, 
is the intersection of its diagonals. Prove easily that 
.
PP21.
is an inscribed quadrilateral . Through 
and 
had a circle tangent to 
at 
; through and 
had a circle tangent to
at 
; through and 
had a circle tangent to at 
; through and had a circle tangent to at . Prove that 
.
Proof. Let
. Easy to see the bisector of 
and the bisector of 
are perpendicular 
. Since
the triangles 
and 
are isosceles 
.
PP22. Let with incircle which touches at , 
and 
. Let 
. Prove that
and 
, where 
are the lengths of the inradii of 
.
Proof.
. Thus,
. In conclusion, 
, what is truly.
Remark. Denote
. Thus, 
. Using the Brianchon's theorem obtain that 
.
PP23. Let and the midpoint of ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, Nagel's point 
, Gergonne's point 
and Feuerbach's point . Prove that 
.
Proof. (See here). Let the second intersection
of the Euler's circle 
with the -median 
, the orthocenter , 
and the midpoint of ![$[AH]$](//latex.artofproblemsolving.com/0/3/b/03b8986ebe750b377f987f87b41a1dbc4c128e17.png)
. Is well-known
and from the power of w.r.t. obtain that 
. Thus, 
, i.e. 
.
, where 
Let 
, where 
From 
and 
obtain that
Thus, 
, i.e. ![$MS=\frac {2m_a\left[a(b+c)-a^2-bc\right]}{b^2+c^2-a^2}\ .$](//latex.artofproblemsolving.com/1/4/5/1456270a82e4a0e97c6abe4b53882cc71d4bd141.png)
Also,
![$\frac {2m_a\left[a(b+c)-a^2-bc\right]}{b^2+c^2-a^2}\cdot \frac {b^2+c^2}{4m_a}$](//latex.artofproblemsolving.com/7/8/7/787f51426e621fd0472a4fd7752bcfdf9cf79bf8.png)
, i.e. 
![$2\left(b^2+c^2\right)\left[a(b+c)-a^2-bc\right]\iff$](//latex.artofproblemsolving.com/e/a/f/eaf760aa5d33554d2eb2d8f82416ba7f2332898c.png)
Observatie. Apropo de subiectul topicului "Puncte importante intr-un triunghi". Pentru cei care nu cunosc semnificatiile punctelor remarcabile mentionate in enuntul problemei propuse
le recomand sa consulte Google, unde, spre surprinderea lor, vor intalni cel putin 100 de asemenea puncte ceea ce inseamna ca triunghiul a fost si ramane o preocupare de secole
a multor matematicieni, multi dintre ei cu rezultate remarcabile la nivel inalt. In demonstratie am folosit urmatoarele relatii metrice dintr-un triunghi care se pot dovedi fara dificultate
and 
PP24. Let a square and its interior point 
for which 
, where 
, i.e. 
and 
. Prove that 
.
Proof. Let
and suppose w.l.o.g. 
. Apply theorem of Sines in 
.
and 
be two circles so that 
The line 
is a common tangent for 
and 
in 
and 
.Let
be the semicircle with 
so that the circles 
are interior tangent to 
Prove that 
Proof. Apply the Pythagoras's theorem to
Therefore,
, where 
Thus, 
Let 
and 
so that 
and 
Observe that 
Apply the Pythagoras'stheorem to
From the relations 
and the relation
obtain that 
From the relations 
and 
obtain that 
. Drawing of PP15 & PP16.PP16. Let
and be two circles so that The line is a common tangent for and in and .Let
be the semicircle with and what is exterior tangent to the circles 
Prove that 
Proof. Apply the Pythagoras's theorem to
Therefore,
Let 
and 
so that and Observe that 
Apply the Pythagoras'stheorem to
From the relations and the relation
obtain that 
From the relations and obtain that 
PP17. Let incircle
of 
and 
, 
so that 
is tangent to 
. Prove that there is the relation 
Proof. Denote
, 
and 
. Thus, ![$[AMN]=2\cdot [AIM]+2\cdot [AIN]-[AEIF]$](http://latex.artofproblemsolving.com/2/0/b/20b2b442ad84abc6a9cd4f54c0881b31e842be7c.png)
, i.e. 
because
and 
. Hence 
, i.e. 
. Thus, 
.The required relation is equivalently with
, i.e. the relation 
.PP18. Let the incircle
of , , and for ![$P\in \overarc[]{EF}$](http://latex.artofproblemsolving.com/5/2/a/52aad01ba7b62fda7f1615ada91abf33dfb7627a.png)
let the lengths 
of the distancies from
to 
, 
, respectively. Prove that 
. Proof. Denote
so that 
and 
so that 
. Thus, 
. Obtain analogously 
. Prove easily that 
. Apply the Ptolemy theorem to the cyclical quadrilateral
.PP19. Let
and 
so that 
. Prove that 
Particularly 
Proof 1. Apply Stewart's relation to cevian
in 
Using an well-knownproperty in
obtain that 
If 
then from obtain that
![$a\cdot MC^2+\left[\left(c^2+ac-b^2\right)-a^2\right]\cdot MC-a\left(c^2+ac-b^2\right)=0$](http://latex.artofproblemsolving.com/1/4/1/141080b68876611a38498bbe09fb2bb4d2203d8d.png)
![$\left(MC-a\right)\left[aMC+c(a+c)-b^2\right]=0\iff$](http://latex.artofproblemsolving.com/9/8/3/9838ae54547e2985ca3e2e4ab314d8416a2b921a.png)
PP20. Let
with the orthocenter 
and the circumcircle 
. Denote
and 
. Prove that 
.Proof. Suppose w.l.o.g. that
. Let 
so that 
is a diameter of 
and 
so that 
. Is well-knownthat
is a parallelogram. Thus, 
is the intersection of its diagonals. Prove easily that 
.PP21.
is an inscribed quadrilateral . Through 
and 
had a circle tangent to 
at 
; through and 
had a circle tangent to
at 
; through and 
had a circle tangent to at 
; through and had a circle tangent to at . Prove that 
.Proof. Let
. Easy to see the bisector of 
and the bisector of 
are perpendicular 
. Since
the triangles 
and 
are isosceles 
.PP22. Let
with incircle which touches at , 
and 
. Let 
. Prove that
and 
, where 
are the lengths of the inradii of 
.Proof.
. Thus,
. In conclusion, 
, what is truly.Remark. Denote
. Thus, 
. Using the Brianchon's theorem obtain that 
.PP23. Let
and the midpoint of ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, Nagel's point 
, Gergonne's point 
and Feuerbach's point . Prove that 
.Proof. (See here). Let the second intersection
of the Euler's circle 
with the -median 
, the orthocenter , 
and the midpoint of ![$[AH]$](http://latex.artofproblemsolving.com/0/3/b/03b8986ebe750b377f987f87b41a1dbc4c128e17.png)
. Is well-known
and from the power of w.r.t. obtain that 
. Thus, 
, i.e. 
.
, where 
Let 
, where 
From 
and 
obtain that
Thus, 
, i.e. ![$MS=\frac {2m_a\left[a(b+c)-a^2-bc\right]}{b^2+c^2-a^2}\ .$](http://latex.artofproblemsolving.com/1/4/5/1456270a82e4a0e97c6abe4b53882cc71d4bd141.png)
Also,
![$\frac {2m_a\left[a(b+c)-a^2-bc\right]}{b^2+c^2-a^2}\cdot \frac {b^2+c^2}{4m_a}$](http://latex.artofproblemsolving.com/7/8/7/787f51426e621fd0472a4fd7752bcfdf9cf79bf8.png)
, i.e. 
![$2\left(b^2+c^2\right)\left[a(b+c)-a^2-bc\right]\iff$](http://latex.artofproblemsolving.com/e/a/f/eaf760aa5d33554d2eb2d8f82416ba7f2332898c.png)
Observatie. Apropo de subiectul topicului "Puncte importante intr-un triunghi". Pentru cei care nu cunosc semnificatiile punctelor remarcabile mentionate in enuntul problemei propuse
le recomand sa consulte Google, unde, spre surprinderea lor, vor intalni cel putin 100 de asemenea puncte ceea ce inseamna ca triunghiul a fost si ramane o preocupare de secole
a multor matematicieni, multi dintre ei cu rezultate remarcabile la nivel inalt. In demonstratie am folosit urmatoarele relatii metrice dintr-un triunghi care se pot dovedi fara dificultate
and 
PP24. Let a square
and its interior point 
for which 
, where 
, i.e. 
and 
. Prove that 
.Proof. Let
and suppose w.l.o.g. 
. Apply theorem of Sines in 
.This post has been edited 175 times. Last edited by Virgil Nicula, Jun 4, 2016, 3:21 PM
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