150. Some (5) problems with fixed points.
by Virgil Nicula, Oct 7, 2010, 8:42 PM
PP1 ("problem of treasury"). Given are two points 
, 
and three mobile points 
, 
, 
so that the line 
doesn't separate , ,
, the triangles 
, 
are isosceles and 
and 
. Prove that the line 
pass through a fixed point.
Proof. Let
with affix 
. Thus, for 
, 
, 
, 
and 
get 
, i.e. the midpoint 
of ![$[MP]$](//latex.artofproblemsolving.com/4/8/2/4821ed42ce01e14fb61be739cc547c042e1a9005.png)
is
fixed, more exactly
is -rightangled and isosceles because 
and 
, where 
is the midpoint of ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
.
PP2. Are given a circle
and a line 
so that 
. Denote the point 
for which 
. For a mobile point 
the circle
with diameter![$[AM]$](//latex.artofproblemsolving.com/1/f/9/1f9b22599237fb6240a50b5f75e8f6ced1292374.png)
cut again the circle 
and the line in the points 
, 
respectively. Prove that the line 
pass through a fixed point (Toshio Seimiya).
Proof. Denote
and the intersection 
between 
and the tangent line in to the circle 
with diameter . Since 
and
, obtain that 
, i.e. the quadrilateral 
is inscribed in the circle with the diameter ![$[AL]$](//latex.artofproblemsolving.com/7/3/7/737a9b49658cbe239832952bb575c8bbdf7dbb35.png)
.
Therefore,
, i.e. 
. Consider the circles , and the null circle (with center and the null radius). Prove easily that the point is the radical center for these three
circles. Since the circle and a point (null circle) are given obtain that the point is fixed and 
.
PP3. Are given a circle
and a line so that . Consider a mobile point . Denote the tangent 
to so that 
and doesn't separate 
, . The tangents from to cut in , . Prove that the -median of 
pass through a fixed point (V.N.).
Proof. Denote the midpoint of ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, the point 
so that 
, 
, 
and 
. Using the standard notations for
obtain
, 
and 
(constant). Since 
is constant obtain that 
is constant, i.e. the point is fixed.
PP4. Let
be a triangle with the circumcircle 
. Given are two fixed points 
, so that separates , and 
separates , .
Consider a mobile point for which 
separates , . Denote 
and 
. Prove that pass through a fixed point (V.N.).
Proof. The Pascal's theorem to the cyclical hexagon
the points 
are collinearly 
pass through the fixed point 
.
PP5. Are given a circle and a line so that . Denote 
for which 
. For a mobile point 
define
the tangent points
of the tangent lines from to so that 
separates , . Denote the projections , 
from on the
tangents
, 
respectively. Prove that the line pass through a fixed point and the line 
pass through a fixed point (O.I.M. - 1995).
Proof. Denote
, 
for which 
, 
, 
and 
. 
is inscribed in the circle with diameter .
Thus
, i.e. is fixed (the pole of w.r.t. ). Thus, 
, , , , belong to circle with diameter ![$[OM]$](//latex.artofproblemsolving.com/6/8/2/682551ce281184bb5ec5b6ffd73d3add53f0ac96.png)
. Since
belongs to circumcircle of
and , , are projections of on the sidelines of , from the Simson's theorem obtain that 
. Observe that 
is inscribed in the circle with the diameter![$[AE]$](//latex.artofproblemsolving.com/0/e/e/0ee81ab9421e97824f7785dc2c44c0f5b04a03b1.png)
and 
is inscribed in the circle with the diameter and 
. Therefore, 
, i.e. in the -right triangle 
have 
. In conclusion, is the midpoint of ![$[EP]$](//latex.artofproblemsolving.com/4/6/1/461402e53a5d8605ce007c50fdc0bd8aaad952f5.png)
,
where , are fixed. Thus and is fixed. More exactly, and from 
obtain that 
.
, 
and three mobile points 
, 
, 
so that the line 
doesn't separate , , , the triangles 
, 
are isosceles and 
and 
. Prove that the line 
pass through a fixed point. Proof. Let
with affix 
. Thus, for 
, 
, 
, 
and 
get 
, i.e. the midpoint 
of ![$[MP]$](http://latex.artofproblemsolving.com/4/8/2/4821ed42ce01e14fb61be739cc547c042e1a9005.png)
isfixed, more exactly
is -rightangled and isosceles because 
and 
, where 
is the midpoint of ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
. PP2. Are given a circle
and a line 
so that 
. Denote the point 
for which 
. For a mobile point 
the circlewith diameter
![$[AM]$](http://latex.artofproblemsolving.com/1/f/9/1f9b22599237fb6240a50b5f75e8f6ced1292374.png)
cut again the circle 
and the line in the points 
, 
respectively. Prove that the line 
pass through a fixed point (Toshio Seimiya).Proof. Denote
and the intersection 
between 
and the tangent line in to the circle 
with diameter . Since 
and
, obtain that 
, i.e. the quadrilateral 
is inscribed in the circle with the diameter ![$[AL]$](http://latex.artofproblemsolving.com/7/3/7/737a9b49658cbe239832952bb575c8bbdf7dbb35.png)
.Therefore,
, i.e. 
. Consider the circles , and the null circle (with center and the null radius). Prove easily that the point is the radical center for these threecircles. Since the circle
and a point (null circle) are given obtain that the point is fixed and 
.PP3. Are given a circle
and a line so that . Consider a mobile point . Denote the tangent 
to so that 
and
doesn't separate 
, . The tangents from to cut in , . Prove that the -median of 
pass through a fixed point (V.N.). Proof. Denote the midpoint
of ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, the point 
so that 
, 
, 
and 
. Using the standard notations for obtain
, 
and 
(constant). Since 
is constant obtain that 
is constant, i.e. the point is fixed. PP4. Let
be a triangle with the circumcircle 
. Given are two fixed points 
, so that separates , and 
separates , .Consider a mobile point
for which 
separates , . Denote 
and 
. Prove that pass through a fixed point (V.N.).Proof. The Pascal's theorem to the cyclical hexagon
the points 
are collinearly 
pass through the fixed point 
. PP5. Are given a circle
and a line so that . Denote 
for which 
. For a mobile point 
definethe tangent points
of the tangent lines from to so that 
separates , . Denote the projections , 
from on thetangents
, 
respectively. Prove that the line pass through a fixed point and the line 
pass through a fixed point (O.I.M. - 1995). Proof. Denote
, 
for which 
, 
, 
and 
. 
is inscribed in the circle with diameter .Thus
, i.e. is fixed (the pole of w.r.t. ). Thus, 
, , , , belong to circle with diameter ![$[OM]$](http://latex.artofproblemsolving.com/6/8/2/682551ce281184bb5ec5b6ffd73d3add53f0ac96.png)
. Since belongs to circumcircle of
and , , are projections of on the sidelines of , from the Simson's theorem obtain that 
. Observe that 
is inscribed in the circle with the diameter
![$[AE]$](http://latex.artofproblemsolving.com/0/e/e/0ee81ab9421e97824f7785dc2c44c0f5b04a03b1.png)
and 
is inscribed in the circle with the diameter and 
. Therefore, 
, i.e. in the -right triangle 
have 
. In conclusion, is the midpoint of ![$[EP]$](http://latex.artofproblemsolving.com/4/6/1/461402e53a5d8605ce007c50fdc0bd8aaad952f5.png)
,where
, are fixed. Thus and is fixed. More exactly, and from 
obtain that 
.This post has been edited 67 times. Last edited by Virgil Nicula, Dec 1, 2015, 10:38 AM
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