31.Some nice metrical problems.
by Virgil Nicula, Apr 26, 2010, 11:36 PM
PP1.
Remark. Can prove easily these relations. If
is the circumcircle of 
, then power 
of 
w.r.t. 
is given by 
a.s.o.
Application. Let an equilateral
with the circumcircle and 
. For 
so that 
separates 
and denote 
and 
. Prove that 
.
Proof. Ptolemy's theorem to
. i.e. 
. Generalized Pythagoras' theorem: 
.
Using upper property
obtain that 
. Thus, 
. Obtain analogously 
Observe
that
and 
implies 
. Thus, 
.
PP2 (Ruben Auqui Dario). Let a
-isosceles with 
and 
. Prove 
.
Proof 1. Let
; 
and 
. Apply the Ptolemy's theorem to 
. Since 
obtain that 
.
Proof 2. Let
. Thus, 
and 
.
Proof 3.
and 
a.s.o.
PP3. Let be a quadrilateral and 
. Prove that 
.
Method I (with the Thales' theorem).
.
Method II (with areas). I"ll use the following simple property : "Let
be two fixed points and 
be two mobile points so that the line 
doesn't separate and
![$[AXB]=[AYB]\ .$](//latex.artofproblemsolving.com/0/2/d/02deefcab4ad6432940f4f1ecfdb3c06ed3c8733.png)
Then 
" . Let 
![$\odot\begin{array}{ccccc}
\nearrow & ABMD\ (BM\parallel AD) & \implies & [AIB]=[DIM] & \searrow\\\\
\searrow & ABCN\ (AN\parallel BC) & \implies & [AIB]=[CIN] & \nearrow\end{array}\odot $](//latex.artofproblemsolving.com/d/3/e/d3e2a0fc62f20f0f6bb8e2678a4cab09ef5e1b89.png)
![$[DIM]=[CIN]\implies$](//latex.artofproblemsolving.com/b/b/d/bbd1ca7e725459b34cd64e40d00f422846a4e7df.png)
![$[DKN]=[CMK]\implies$](//latex.artofproblemsolving.com/0/2/0/02074ceeae96aa6188f9bc281c1682cdc2b19cbb.png)
![$[CND]=[CMD]\implies MN\parallel CD$](//latex.artofproblemsolving.com/d/9/7/d97e505cd225fddacf317a0b3f8d8c40fe7a17ac.png)
Method III (with trigonometry). Denote
.
PP4. Let
be a line and for 
construct two similar triangles 
so that the line
doesn't separate
. Denote 
so that 
and 
. Prove that 
.
Proof 1. Let
and 
where 
, i.e. 
. Thus, 
.
Proof 2.
.
PP5 (Ruben Dario). Let an-right with incircle 
which touches , 
at 
, 
. Let rectangle 
so that
and 
. Prove that 
, i.e. 
.
Proof. Denote the length
of the -altitude, i.e. 
. Observe that 
, i.e. 
. Denote the projection 
, 
of , on 
. So 
.
PP6 (Cristian Tello). Let:- right 
so that 
so that 
the exterior squares and 
with diameter ![$[UA]$](//latex.artofproblemsolving.com/4/6/0/4605298b2579f7182d5c23aaa6a05654402b838a.png)
and
with diameter ![$[UE]\ ;$](//latex.artofproblemsolving.com/f/3/0/f304a1de6cc3e4fdc0f0afc704c45d7a27ea53d6.png)
the common (exterior) tangent HJ of 
, 
, where 
, 
the tangent 
to from 
, where 
and the tangent 
to from
, where 
so that 
and 
separates and 
the midpoint 
of ![$[DF]$](//latex.artofproblemsolving.com/4/8/7/487608ba746e637d846b20401f23cc2b80336338.png)
. Prove that 
.
Proof. Let
. Observe that 
and 
. Thus, 
. From 
obtain that 
i.e.
. On other hand, 
Otherwise. Let
. I"ll use the power of 
.
Theorem of median in
![$2\left(a^2+b^2\right)+2h(a+b)-\left[(b+a)^2-(b-a)^2\right]=2h(a+b)\implies$](//latex.artofproblemsolving.com/a/3/6/a3657c51fa38d57d646737db5aace16c83e1e74c.png)
. In conclusion, from 
and 
get 
, i.e. 
.
PP7 (Miguel Ochoa Sanchez). Let a cyclic for which suppose that exists an interior 
so that 
. Prove 
.
Proof (Rubén HG). Let
. Thus, 
.
Prove easily
and I"ll use identity 
. Apply the theorem of Sines in the triangles 
.
.
Remark. Can prove easily these relations. If
is the circumcircle of 
, then power 
of 
w.r.t. 
is given by 
a.s.o.Application. Let an equilateral
with the circumcircle and 
. For 
so that 
separates 
and denote 
and 
. Prove that 
.Proof. Ptolemy's theorem to
. i.e. 
. Generalized Pythagoras' theorem: 
.Using upper property
obtain that 
. Thus, 
. Obtain analogously 
Observethat
and 
implies 
. Thus, 
.PP2 (Ruben Auqui Dario). Let a
-isosceles with 
and 
. Prove 
. Proof 1. Let
; 
and 
. Apply the Ptolemy's theorem to 
. Since 
obtain that 
.Proof 2. Let
. Thus, 
and 
.Proof 3.
and 
a.s.o.PP3. Let
be a quadrilateral and 
. Prove that 
.Method I (with the Thales' theorem).
.Method II (with areas). I"ll use the following simple property : "Let
be two fixed points and 
be two mobile points so that the line 
doesn't separate and![$[AXB]=[AYB]\ .$](http://latex.artofproblemsolving.com/0/2/d/02deefcab4ad6432940f4f1ecfdb3c06ed3c8733.png)
Then 
" . Let 
![$\odot\begin{array}{ccccc}
\nearrow & ABMD\ (BM\parallel AD) & \implies & [AIB]=[DIM] & \searrow\\\\
\searrow & ABCN\ (AN\parallel BC) & \implies & [AIB]=[CIN] & \nearrow\end{array}\odot $](http://latex.artofproblemsolving.com/d/3/e/d3e2a0fc62f20f0f6bb8e2678a4cab09ef5e1b89.png)
![$[DIM]=[CIN]\implies$](http://latex.artofproblemsolving.com/b/b/d/bbd1ca7e725459b34cd64e40d00f422846a4e7df.png)
![$[DKN]=[CMK]\implies$](http://latex.artofproblemsolving.com/0/2/0/02074ceeae96aa6188f9bc281c1682cdc2b19cbb.png)
![$[CND]=[CMD]\implies MN\parallel CD$](http://latex.artofproblemsolving.com/d/9/7/d97e505cd225fddacf317a0b3f8d8c40fe7a17ac.png)
Method III (with trigonometry). Denote
.PP4. Let
be a line and for 
construct two similar triangles 
so that the line doesn't separate
. Denote 
so that 
and 
. Prove that 
.Proof 1. Let
and 
where 
, i.e. 
. Thus, 
.Proof 2.
.PP5 (Ruben Dario). Let an
-right with incircle 
which touches , 
at 
, 
. Let rectangle 
so that
and 
. Prove that 
, i.e. 
.Proof. Denote the length
of the -altitude, i.e. 
. Observe that 
, i.e. 
. Denote the projection 
, 
of , on 
. So 
.PP6 (Cristian Tello). Let:
- right 
so that 
so that 
the exterior squares and 
with diameter ![$[UA]$](http://latex.artofproblemsolving.com/4/6/0/4605298b2579f7182d5c23aaa6a05654402b838a.png)
and
with diameter ![$[UE]\ ;$](http://latex.artofproblemsolving.com/f/3/0/f304a1de6cc3e4fdc0f0afc704c45d7a27ea53d6.png)
the common (exterior) tangent HJ of 
, 
, where 
, 
the tangent 
to from 
, where 
and the tangent 
to from , where 
so that 
and 
separates and 
the midpoint 
of ![$[DF]$](http://latex.artofproblemsolving.com/4/8/7/487608ba746e637d846b20401f23cc2b80336338.png)
. Prove that 
. Proof. Let
. Observe that 
and 
. Thus, 
. From 
obtain that 
i.e.
. On other hand, 
Otherwise. Let
. I"ll use the power of 
.Theorem of median in
![$2\left(a^2+b^2\right)+2h(a+b)-\left[(b+a)^2-(b-a)^2\right]=2h(a+b)\implies$](http://latex.artofproblemsolving.com/a/3/6/a3657c51fa38d57d646737db5aace16c83e1e74c.png)
. In conclusion, from 
and 
get 
, i.e. 
.PP7 (Miguel Ochoa Sanchez). Let a cyclic
for which suppose that exists an interior 
so that 
. Prove 
.Proof (Rubén HG). Let
. Thus, 
.Prove easily
and I"ll use identity 
. Apply the theorem of Sines in the triangles 
.
.
This post has been edited 321 times. Last edited by Virgil Nicula, Nov 22, 2016, 2:49 PM
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