333. Problems with the common tangent for two circles.
by Virgil Nicula, Jan 17, 2012, 10:58 AM
PP1. Let 
, 
and 
be three circles so that 
and 
are externally tangent and both are internally tangent to 
.
Find the length 
of the chord from the large circle which is the interior tangent to the small circles (at its common tangent point).
Find the length 
of the chord from the larger circle which is the exterior tangent to the small circles (at two points) - two cases.
Proof. Denote
so that 
and 
so that 
is common tangent of and 
so that 
.
Apply the power of 
w.r.t. 
. Observe that 
and 
.
Apply the Stewart's theorem to the cevian
in 
. Using the relation 
obtain that 
. Observe that 
. Apply the Casey's theorem
to the circles , and the degenerated circles 
, 
. Obtain that 
.
Appear two cases : the first case when 
separates 
and the second case when doesn't separate . Denote 
, 
.
From an well-known property the rays
, 
are the bisectors of the angles 
, 
respectively 
belongs to and it is the
midpoint of the arc
. Apply the Casey's theorem to the circles , and the degenerated circles 
, 
. Obtain that 
what is the evident relation 
. Soon !!!
PP2. Let
be a triangle with the incircle 
and the circumcircle . Consider the circle 
which is tangent to 
, 
and which
is internal tangent to . Denote 
. The tangent from 
to touches in . Prove that 
.
Proof. Assume w.l.o.g. that
and let 
and 
be the tangency points of 
with and respectively. By Casey's theorem for the circle and
the degenerated circles , , all tangent to the circumcircle 
we get 
. By Ptolemy's theorem for 
.
PP3 (Balkan MO 2012). Let be an acute-angled triangle with orthocentre 
and orthic triangle 
, where 
, 
, 
. Suppose that
meets the circle with diameter ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
at 
. Show that the circumcircles of 
and 
are tangent at 
(this problem is modified equivalent).
Proof 1 (metric). In the right-angled 
from the theorem of cathet ![$[PC]$](//latex.artofproblemsolving.com/7/8/1/7818452d29e34a743729cef6d6762490b71f626d.png)
obtain that 
. Since 
, 
are cyclical quadrilaterals obtain
that
. In conclusion, 
, i.e. the circumcircles of and are tangent at .
Proof 2 (synthetic). Prove easily that
PP4. Let an acute
with 
. Let be the orthocenter of triangle and let be the midpoint of the side 
.
Let be a point on the side and 
a point on the side such that 
and the points , , are on the same line.
Prove that the line
is perpendicular to the common chord of the circumscribed circles of triangle and triangle 
.
Lemma. Let be an acute triangle. Define: the circumcircle 
and the orthocentre of the triangle ; the middlepoint of the side ; the intersection between
and the bisector of the angle 
; 
and 
so that 
and 
. Then the point belongs to the circumcircle 
of the triangle 
.
Proof of the lemma.
and 
. Thus, 
Denote the point 
, i.e. 
. Thus, 
and 
, i.e. belongs to the circumcircle of 
Proof of the proposed problem. Denote:
; the middlepoint 
![$[AH]$](//latex.artofproblemsolving.com/0/3/b/03b8986ebe750b377f987f87b41a1dbc4c128e17.png)
; 
. From above lemma results 
.
But
( is the middlepoint of the segment ![$[HA']$](//latex.artofproblemsolving.com/b/d/d/bdd6284b56db3edb70c48d53b2cef3bbf7dfdb34.png)
), 
and 
. Thus , 
, 
are the middlepoints of
, 
, 
respectively and 
and 
. Thus, 
. Remark. I denote ray 
without 
.
, 
and 
be three circles so that 
and 
are externally tangent and both are internally tangent to 
.
Find the length 
of the chord from the large circle which is the interior tangent to the small circles (at its common tangent point).
Find the length 
of the chord from the larger circle which is the exterior tangent to the small circles (at two points) - two cases.Proof. Denote
so that 
and 
so that 
is common tangent of and 
so that 
. Apply the power of 
w.r.t. 
. Observe that 
and 
.Apply the Stewart's theorem to the cevian
in 
. Using the relation 
obtain that 
. Observe that 
. Apply the Casey's theoremto the circles
, and the degenerated circles 
, 
. Obtain that 
. Appear two cases : the first case when 
separates 
and the second case when doesn't separate . Denote 
, 
.From an well-known property the rays
, 
are the bisectors of the angles 
, 
respectively 
belongs to and it is themidpoint of the arc
. Apply the Casey's theorem to the circles , and the degenerated circles 
, 
. Obtain that 
what is the evident relation 
. Soon !!!PP2. Let
be a triangle with the incircle 
and the circumcircle . Consider the circle 
which is tangent to 
, 
and whichis internal tangent to
. Denote 
. The tangent from 
to touches in . Prove that 
.Proof. Assume w.l.o.g. that
and let 
and 
be the tangency points of 
with and respectively. By Casey's theorem for the circle andthe degenerated circles
, , all tangent to the circumcircle 
we get 
. By Ptolemy's theorem for 
.PP3 (Balkan MO 2012). Let
be an acute-angled triangle with orthocentre 
and orthic triangle 
, where 
, 
, 
. Suppose that
meets the circle with diameter ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
at 
. Show that the circumcircles of 
and 
are tangent at 
(this problem is modified equivalent).Proof 1 (metric). In the
right-angled 
from the theorem of cathet ![$[PC]$](http://latex.artofproblemsolving.com/7/8/1/7818452d29e34a743729cef6d6762490b71f626d.png)
obtain that 
. Since 
, 
are cyclical quadrilaterals obtainthat
. In conclusion, 
, i.e. the circumcircles of and are tangent at .Proof 2 (synthetic). Prove easily that
PP4. Let an acute
with 
. Let be the orthocenter of triangle and let be the midpoint of the side 
.Let
be a point on the side and 
a point on the side such that 
and the points , , are on the same line.Prove that the line
is perpendicular to the common chord of the circumscribed circles of triangle and triangle 
.Lemma. Let
be an acute triangle. Define: the circumcircle 
and the orthocentre of the triangle ; the middlepoint of the side ; the intersection between
and the bisector of the angle 
; 
and 
so that 
and 
. Then the point belongs to the circumcircle 
of the triangle 
.Proof of the lemma.
and 
. Thus, 
Denote the point 
, i.e. 
. Thus, 
and 
, i.e. belongs to the circumcircle of 
Proof of the proposed problem. Denote:
; the middlepoint 
![$[AH]$](http://latex.artofproblemsolving.com/0/3/b/03b8986ebe750b377f987f87b41a1dbc4c128e17.png)
; 
. From above lemma results 
.But
( is the middlepoint of the segment ![$[HA']$](http://latex.artofproblemsolving.com/b/d/d/bdd6284b56db3edb70c48d53b2cef3bbf7dfdb34.png)
), 
and 
. Thus , 
, 
are the middlepoints of, 
, 
respectively and 
and 
. Thus, 
. Remark. I denote ray 
without 
.This post has been edited 77 times. Last edited by Virgil Nicula, Nov 19, 2015, 1:20 PM
August 2018
January 2018