25. Geometric equations.
by Virgil Nicula, Apr 22, 2010, 10:29 PM
PP1. Let 
be triangle. Denote the distance 
from the point 
to the side 
and 
. Prove that :
PP2. Let
a real number and the triangle . Prove that there is the following identity: 
.
Study some particular cases, for example
. Solve the ecuation: 
, where 
.
PP3. Let be an equilateral triangle with 
. Consider two points 
, 
so that 
and 
is a tangential
quadrilateral, where
. Prove that if the 
and 
have same inradius 
, then 
. Nice !
Proof 1. Denote the incenters
and 
of 
respectively. Prove easily that 
, 
, 
and 
.
Observe that
and apply the theorem of Cosinus in 
. Obtain that 
![$d^2=\left(1+r^2\right)+\left[1+(r+d)^2\right]-\sqrt{3\left(1+r^2\right)\left[1+(r+d)^2\right]}\iff$](//latex.artofproblemsolving.com/7/a/c/7ac1707dae298d26f01f502003b0f03a7abace55.png)
![$2\left(1+r^2+rd\right)=\sqrt{3\left(1+r^2\right)\left[1+(r+d)^2\right]}\iff$](//latex.artofproblemsolving.com/1/6/4/1645caa62eecc5296c222a2c424b799569af25e3.png)
Now plug in 
. Obviously can't exceed the altitude of the triangle, which is 
. Hence 
.
Proof 2. Denote the midpoint
of ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, the incenters and of respectively and 
. Prove easily that ,
and
. Thus, 
![$\tan\left[\left(60^{\circ}-x\right)-\left(30^{\circ}-x\right)\right]=\frac {\left(\sqrt 3-2r\right)-r}{1+r\left(\sqrt 3-2r\right)}\implies$](//latex.artofproblemsolving.com/8/a/f/8afa8e237b1c27f40fe0e111a2f4342d2dc31084.png)
.
An easy extension. Let an-isosceles 
with and , so that and is a tangential
quadrilateral, where . Prove that and have same inradius 
.
be triangle. Denote the distance 
from the point 
to the side 
and 
. Prove that :
PP2. Let
a real number and the triangle . Prove that there is the following identity: 
.Study some particular cases, for example
. Solve the ecuation: 
, where 
.PP3. Let
be an equilateral triangle with 
. Consider two points 
, 
so that 
and 
is a tangentialquadrilateral, where
. Prove that if the 
and 
have same inradius 
, then 
. Nice ! Proof 1. Denote the incenters
and 
of 
respectively. Prove easily that 
, 
, 
and 
.Observe that
and apply the theorem of Cosinus in 
. Obtain that 
![$d^2=\left(1+r^2\right)+\left[1+(r+d)^2\right]-\sqrt{3\left(1+r^2\right)\left[1+(r+d)^2\right]}\iff$](http://latex.artofproblemsolving.com/7/a/c/7ac1707dae298d26f01f502003b0f03a7abace55.png)
![$2\left(1+r^2+rd\right)=\sqrt{3\left(1+r^2\right)\left[1+(r+d)^2\right]}\iff$](http://latex.artofproblemsolving.com/1/6/4/1645caa62eecc5296c222a2c424b799569af25e3.png)
Now plug in 
. Obviously can't exceed the altitude of the triangle, which is 
. Hence 
.Proof 2. Denote the midpoint
of ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, the incenters and of respectively and 
. Prove easily that , and
. Thus, 
![$\tan\left[\left(60^{\circ}-x\right)-\left(30^{\circ}-x\right)\right]=\frac {\left(\sqrt 3-2r\right)-r}{1+r\left(\sqrt 3-2r\right)}\implies$](http://latex.artofproblemsolving.com/8/a/f/8afa8e237b1c27f40fe0e111a2f4342d2dc31084.png)
.An easy extension. Let an
-isosceles 
with and , so that and is a tangentialquadrilateral, where
. Prove that and have same inradius 
.This post has been edited 31 times. Last edited by Virgil Nicula, May 6, 2016, 11:26 PM
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