93. The Appolonius' construct problems.
by Virgil Nicula, Aug 28, 2010, 7:57 PM
Lemma P.P.L. Given are two points 
, 
and a line 
such that 
, the line doesn't
separate , and 
. Construct the circles which are tangent to and pass through , .
Proof. Denote
and suppose w.l.o.g. 
. Define the tangent point 
of the required circle 
.
Observe that
belongs to the bisector of the segment ![$ [AB]$](//latex.artofproblemsolving.com/d/7/a/d7a8027c238eec9cf67de0f7ec6cb1df4df49a61.png)
and 
. Consider the circle 
with diameter ![$ [PB]$](//latex.artofproblemsolving.com/a/a/8/aa8ee3823f0fbb815fbbfeba11a8b1ade67d6086.png)
and 
for which
. Observe that 
, i.e. 
. Thus, 
belongs to the circle with center 
and radius![$ [PC]$](//latex.artofproblemsolving.com/5/b/5/5b579053a08a07ea5240ba4cb2f437d1c751844f.png)
. Appear just two solutions. Now the construction of the required circles is easily.
Lemma P.P.C. Given are two points , and a circle 
such that 
. Construct the circles which are tangent to and 
.
Proof. Construct a circle
which pass through , and is secant to (in the points 
, 
). Construct from 
the tangent lines (in the points
, 
) to the circle . Then the required circles are the circumcircles of 
, 
.
Lemma P.L.C. Given are a point , a line and a circle such that 
, 
and
the line doesn't separate , . Construct the circles which are tangent to , and which pass through .
Proof. Denote the diameter![$ [BC]$](//latex.artofproblemsolving.com/3/5/5/3550468aa97af843ef34b8868728963dec043efe.png)
of for which 
and 
. Suppose w.l.o.g. 
. Define the tangent points 
, 
of the required circle with , respectively. Show easily that 
and the quadrilateral 
is cyclically. Denote the second
intersection
of 
with circle . Observe that 
, i.e. belongs to the line and to the circumcircle
of the triangle
. Now the construction of a circle becomes the construction of a circle which pass through two given points ,
and which is tangent to a given line (see lemma P.P.L.).
Lemma P.C.C. Given are a point , and two circles 
and 
such that 
,
and 
. Construct the circles which pass through and which are exterior tangent to 
, 
.
Proof. Construct the intersection of
with a common exterior tangent 
of , , where 
, 
.
The required circle is exterior tangent to , in , respectively. Show easily that 
and the quadrilateral 
is cyclically. Denote the second intersection of with 
. Therefore, 
, i.e. the point 
and to the circumcircle of
. Now the construction of a circle becomes the construction of a circle which pass through
two given points , and which is tangent to a given circle (or ) (see lemma P.P.C.).
Lemma L.C.C. Given are a line and two circles , such that the line
doesn't separate
, 
and 
. Construct the circles which are exterior tangent to , and .
Proof. Suppose w.l.o.g. that
. Consider the circle 
with center and radius 
. Construct the line 
for which
the distance between , 
is equally to 
and the line "separates" the circles , from the line . Construct the circle 
which pass through and which is tangent to and (see lemma P.L.C.). Then the required circle is 
.
Remark. The construction of the circles which are tangent (exterior or interior) to three given circles, being possibly degenerated
to a line (
) or a point (
) is named the Appolonius' construct problem. Using the symbols P (point), L (line), C (circle)
there are ten Appolonius' construct problems : P.P.P. ; P.P.L. ; P.P.C. ; P.D.D. ; P.D.C. ; P.C.C. ; D.D.D. ; D.D.C. ; D.C.C. ; C.C.C.
In conclusion, the solving of this remarkable construct problem can illustrate on the following sketch of "reduction to previous problem" :
L.L.C.
P.L.L. P.P.L. P.P.P. ; L.C.C. P.L.C. P.P.L. ; C.C.C. P.C.C. P.P.C. P.P.P.
, 
and a line 
such that 
, the line doesn'tseparate
, and 
. Construct the circles which are tangent to and pass through , . Proof. Denote
and suppose w.l.o.g. 
. Define the tangent point 
of the required circle 
.Observe that
belongs to the bisector of the segment ![$ [AB]$](http://latex.artofproblemsolving.com/d/7/a/d7a8027c238eec9cf67de0f7ec6cb1df4df49a61.png)
and 
. Consider the circle 
with diameter ![$ [PB]$](http://latex.artofproblemsolving.com/a/a/8/aa8ee3823f0fbb815fbbfeba11a8b1ade67d6086.png)
and 
for which
. Observe that 
, i.e. 
. Thus, 
belongs to the circle with center 
and radius
![$ [PC]$](http://latex.artofproblemsolving.com/5/b/5/5b579053a08a07ea5240ba4cb2f437d1c751844f.png)
. Appear just two solutions. Now the construction of the required circles is easily. Lemma P.P.C. Given are two points
, and a circle 
such that 
. Construct the circles which are tangent to and 
. Proof. Construct a circle
which pass through , and is secant to (in the points 
, 
). Construct from 
the tangent lines (in the points
, 
) to the circle . Then the required circles are the circumcircles of 
, 
.Lemma P.L.C. Given are a point
, a line and a circle such that 
, 
andthe line
doesn't separate , . Construct the circles which are tangent to , and which pass through . Proof. Denote the diameter
![$ [BC]$](http://latex.artofproblemsolving.com/3/5/5/3550468aa97af843ef34b8868728963dec043efe.png)
of for which 
and 
. Suppose w.l.o.g. 
. Define the tangent points 
, 
of the required circle
with , respectively. Show easily that 
and the quadrilateral 
is cyclically. Denote the secondintersection
of 
with circle . Observe that 
, i.e. belongs to the line and to the circumcircleof the triangle
. Now the construction of a circle becomes the construction of a circle which pass through two given points , and which is tangent to a given line
(see lemma P.P.L.).Lemma P.C.C. Given are a point
, and two circles 
and 
such that 
,
and 
. Construct the circles which pass through and which are exterior tangent to 
, 
. Proof. Construct the intersection of
with a common exterior tangent 
of , , where 
, 
.The required circle
is exterior tangent to , in , respectively. Show easily that 
and the quadrilateral 
is cyclically. Denote the second intersection
of with 
. Therefore, 
, i.e. the point 
and to the circumcircle of
. Now the construction of a circle becomes the construction of a circle which pass throughtwo given points
, and which is tangent to a given circle (or ) (see lemma P.P.C.).Lemma L.C.C. Given are a line
and two circles , such that the line doesn't separate
, 
and 
. Construct the circles which are exterior tangent to , and . Proof. Suppose w.l.o.g. that
. Consider the circle 
with center and radius 
. Construct the line 
for whichthe distance between
, 
is equally to 
and the line "separates" the circles , from the line . Construct the circle 
which pass through
and which is tangent to and (see lemma P.L.C.). Then the required circle is 
.Remark. The construction of the circles which are tangent (exterior or interior) to three given circles, being possibly degenerated
to a line (
) or a point (
) is named the Appolonius' construct problem. Using the symbols P (point), L (line), C (circle)there are ten Appolonius' construct problems : P.P.P. ; P.P.L. ; P.P.C. ; P.D.D. ; P.D.C. ; P.C.C. ; D.D.D. ; D.D.C. ; D.C.C. ; C.C.C.
In conclusion, the solving of this remarkable construct problem can illustrate on the following sketch of "reduction to previous problem" :
L.L.C.
P.L.L. P.P.L. P.P.P. ; L.C.C. P.L.C. P.P.L. ; C.C.C. P.C.C. P.P.C. P.P.P. This post has been edited 3 times. Last edited by Virgil Nicula, Nov 23, 2015, 3:22 PM
August 2018
January 2018