166. Some constant geometrical expressions.
by Virgil Nicula, Oct 25, 2010, 7:40 AM
PP1. Let 
be an 
-isoceles triangle. For a mobile point 
so that 
denote the length of the
inradius
of 
and the 
-exinradius 
of 
. Prove that the sum 
(constant). (Virgil Nicula).
Proof. Observe that
and 
. Apply the Stewart's relation to the cevian ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
in 
. Otherwise. Denote
the midpoint
of ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. Thus, 
, i.e. 
.
Therefore,
, i.e. the relation 
.
PP2. Given are a fixed segment and a fixed line 
so that 
. Denote the circle 
with diameter and suppose that 
. For a mobile
the tangent lines from to 
meet 
in 
, 
. Prove that the ratio 
is constant and the inradius of 
is also constant (Polonia, 1994).
Proof. Since the ray
is the -bisector in 
results that 
. Denote the distance 
, 
for which 
and
. From 
obtain 
. Thus, 
(constant). Denote the incenter 
of .
Then
and 
(constant). Thus, the locus of is a parallel line to , i.e. the length of the inradius of is constant.
Remark. Denote the midpoint of ![$[MN]$](//latex.artofproblemsolving.com/1/6/2/162e03b9cf481dcfb3d5bdf078be84feab5d2f6e.png)
and the point 
so that 
. Observe that 
and
, i.e. 
(constant). In conclusion the -median of
pass through a fixed point
for which and . With other words, the point belongs to the polar of 
w.r.t. , where 
and 
.
PP3. Consider two mobile points
(fixed) so that 
. Denote the points 
for which 
and 
,
. Prove that the expression 
(constant), where 
sau 
if the line separates or doesn't separate 
, (Virgil Nicula).
Proof. Denote
, 
. Observe that in the first case when separates ,
we have
and in the second case when doesn't separate , we have 
. Therefore :
Case I
and 
![$2r^2\left[2-(\cos 2\alpha +\cos 2\beta )+\cos (\alpha -\beta )-\cos (\alpha +\beta )\right]=$](//latex.artofproblemsolving.com/2/3/b/23ba3e7faa472cfe92280449097e856ca638979c.png)
![$2r^2\left[2-2\cos (\alpha +\beta )\cos (\alpha -\beta )-\frac 12-\cos (\alpha +\beta )\right]=$](//latex.artofproblemsolving.com/f/8/b/f8b2b8372d8447db9fa77b380524043aacca145f.png)
![$2r^2\left[\frac 32+\cos (\alpha +\beta )-\cos (\alpha +\beta )\right]=3r^2$](//latex.artofproblemsolving.com/2/f/6/2f666c87dc2aa12ce1b045ea5b836afe0b664cf0.png)
.
Case II
and 
![$2r^2\left[2-(\cos 2\alpha +\cos 2\beta )-\cos (\alpha -\beta )+\cos (\alpha +\beta )\right]=$](//latex.artofproblemsolving.com/1/b/3/1b3895ce65cc433095a74eef8fbfe7453e154ad7.png)
![$2r^2\left[2-2\cos (\alpha +\beta )\cos (\alpha -\beta )-\cos (\alpha -\beta )-\frac 12\right]=$](//latex.artofproblemsolving.com/7/4/0/74028e78aac98bac52f82e866a7609b63261308a.png)
![$2r^2\left[\frac 32+\cos (\alpha -\beta )-\cos (\alpha -\beta )\right]=3r^2$](//latex.artofproblemsolving.com/b/6/8/b686224703c457757844b4da345f77abc78723d3.png)
.
In conclusion, in both cases the corresponding expressions are constant and the constant is the same
.
PP4. Let
with the centroid 
and a fixed 
. Consider a variable circle 
for which 
and its center belongs to a fixed line 
.
Prove that the sum of the powers w.r.t. of , 
, is constant, i.e. exists 
so that 
for any with mentioned properties.
Proof. Since
and exists 
so that 
. Using the well-known property 
obtain
(constant).
PP5. Let with the incircle . Denote the midpoint of and 
, 
. Let the incircles 
of the triangles
, 
respectively. Prove that 
and in this case 
(Toshio Seimyia).
Proof. Denote the points of contact
, 
with 
of 
, 
respectively and the points of contact 
, 
with , 
of , respectively. Observe that
. Since 
results 
. Since 
results
. From the relations 
, 
, 
obtain 
, i.e, 
. Thus 
and
. Therefore, 
. In this case
. Since 
and obtain that 
. Very nice !
Here are some simple problems with constant geometrical expressions (without proofs).
PS1. Let an-isosceles with circumcircle . Prove that 
for which doesn't separate , the expression 
is constant.
PS2. Are given
with diameter ; a fixed ![$D\in [AB]$](//latex.artofproblemsolving.com/1/4/7/14735b3986bc5d9721faf76fc3a97f41f4118775.png)
, a mobile 
; for which 
. Prove that 
is constant.
PS3. Let be a triangle with the circumcircle . For a mobile point 
denote 
, 
. Prove that 
(constant). Very nice !
PS4. Let a cyclic
, where 
, 
and the midpoints , of , ![$[CD]$](//latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
. Prove that 
(Bulgaria, 1997).
be an 
-isoceles triangle. For a mobile point 
so that 
denote the length of theinradius
of 
and the 
-exinradius 
of 
. Prove that the sum 
(constant). (Virgil Nicula). Proof. Observe that
and 
. Apply the Stewart's relation to the cevian ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
in 
. Otherwise. Denotethe midpoint
of ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. Thus, 
, i.e. 
.Therefore,
, i.e. the relation 
.PP2. Given are a fixed segment
and a fixed line 
so that 
. Denote the circle 
with diameter and suppose that 
. For a mobile
the tangent lines from to 
meet 
in 
, 
. Prove that the ratio 
is constant and the inradius of 
is also constant (Polonia, 1994). Proof. Since the ray
is the -bisector in 
results that 
. Denote the distance 
, 
for which 
and
. From 
obtain 
. Thus, 
(constant). Denote the incenter 
of .Then
and 
(constant). Thus, the locus of is a parallel line to , i.e. the length of the inradius of is constant.Remark. Denote the midpoint
of ![$[MN]$](http://latex.artofproblemsolving.com/1/6/2/162e03b9cf481dcfb3d5bdf078be84feab5d2f6e.png)
and the point 
so that 
. Observe that 
and
, i.e. 
(constant). In conclusion the -median of pass through a fixed point
for which and . With other words, the point belongs to the polar of 
w.r.t. , where 
and 
. PP3. Consider two mobile points
(fixed) so that 
. Denote the points 
for which 
and 
,
. Prove that the expression 
(constant), where 
sau 
if the line separates or doesn't separate 
, (Virgil Nicula).Proof. Denote
, 
. Observe that in the first case when separates , we have
and in the second case when doesn't separate , we have 
. Therefore :Case I
and 
![$2r^2\left[2-(\cos 2\alpha +\cos 2\beta )+\cos (\alpha -\beta )-\cos (\alpha +\beta )\right]=$](http://latex.artofproblemsolving.com/2/3/b/23ba3e7faa472cfe92280449097e856ca638979c.png)
![$2r^2\left[2-2\cos (\alpha +\beta )\cos (\alpha -\beta )-\frac 12-\cos (\alpha +\beta )\right]=$](http://latex.artofproblemsolving.com/f/8/b/f8b2b8372d8447db9fa77b380524043aacca145f.png)
![$2r^2\left[\frac 32+\cos (\alpha +\beta )-\cos (\alpha +\beta )\right]=3r^2$](http://latex.artofproblemsolving.com/2/f/6/2f666c87dc2aa12ce1b045ea5b836afe0b664cf0.png)
.Case II
and 
![$2r^2\left[2-(\cos 2\alpha +\cos 2\beta )-\cos (\alpha -\beta )+\cos (\alpha +\beta )\right]=$](http://latex.artofproblemsolving.com/1/b/3/1b3895ce65cc433095a74eef8fbfe7453e154ad7.png)
![$2r^2\left[2-2\cos (\alpha +\beta )\cos (\alpha -\beta )-\cos (\alpha -\beta )-\frac 12\right]=$](http://latex.artofproblemsolving.com/7/4/0/74028e78aac98bac52f82e866a7609b63261308a.png)
![$2r^2\left[\frac 32+\cos (\alpha -\beta )-\cos (\alpha -\beta )\right]=3r^2$](http://latex.artofproblemsolving.com/b/6/8/b686224703c457757844b4da345f77abc78723d3.png)
.In conclusion, in both cases the corresponding expressions are constant and the constant is the same
.PP4. Let
with the centroid 
and a fixed 
. Consider a variable circle 
for which 
and its center belongs to a fixed line 
.Prove that the sum of the powers w.r.t.
of , 
, is constant, i.e. exists 
so that 
for any with mentioned properties. Proof. Since
and exists 
so that 
. Using the well-known property 
obtain
(constant).PP5. Let
with the incircle . Denote the midpoint of and 
, 
. Let the incircles 
of the triangles
, 
respectively. Prove that 
and in this case 
(Toshio Seimyia).Proof. Denote the points of contact
, 
with 
of 
, 
respectively and the points of contact 
, 
with , 
of , respectively. Observe that
. Since 
results 
. Since 
results
. From the relations 
, 
, 
obtain 
, i.e, 
. Thus 
and
. Therefore, 
. In this case
. Since 
and obtain that 
. Very nice !Here are some simple problems with constant geometrical expressions (without proofs).
PS1. Let an
-isosceles with circumcircle . Prove that 
for which doesn't separate , the expression 
is constant.PS2. Are given
with diameter ; a fixed ![$D\in [AB]$](http://latex.artofproblemsolving.com/1/4/7/14735b3986bc5d9721faf76fc3a97f41f4118775.png)
, a mobile 
; for which 
. Prove that 
is constant.PS3. Let
be a triangle with the circumcircle . For a mobile point 
denote 
, 
. Prove that 
(constant). Very nice !PS4. Let a cyclic
, where 
, 
and the midpoints , of , ![$[CD]$](http://latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
. Prove that 
(Bulgaria, 1997).This post has been edited 80 times. Last edited by Virgil Nicula, Dec 1, 2015, 10:04 AM
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