205. A "slicing" problem with 11 methods.
by Virgil Nicula, Jan 6, 2011, 8:05 PM
PP1 (Sergey Yurin, Kvant). Let 
with 
and 
. Consider 
for which 
. Find 
.
• • • • • • • • • • • •
PP2. Let
be a square and consider its interior point 
so that 
. Prove that the triangle 
is an equilateral triangle.
Proof 1. Since
obtain that 
. Denote 
and the projection 
of the point 
on the line 
. Prove easily that 
,
, 
and 
. Therefore, the triangle 
is -isosceles. In conclusion, 
![$\frac 12\cdot\left[180^{\circ}-m\left(\widehat{MFB}\right)\right]=$](//latex.artofproblemsolving.com/b/2/1/b21a1501757b9815237cf7d495f5ae5ba69bf423.png)
, what means the triangle 
is equilateral .
Proof 2. Construct the equilateral
, where 
separates and 
. Observe that 
and the ray 
is the
bisector of
. Thus, 
. Therefore, 
is equilateral.
Proof 3. Denote the projection
of 
on 
.Observe that 
, the quadrilateral 
is inscribed in the circle with the diameter ![$[BD]$](//latex.artofproblemsolving.com/3/2/6/3261e689901bce018ecdef47b9bc60a78ead3746.png)
and 
.
Thus,
is equilateral.
Proof 4. Soon.
with 
and 
. Consider 
for which 
. Find 
.• • • • • • • • • • • •
. Therefore, 
.
so that the line 
doesn't separate 
, 
is
obtain that 
. Therefore, 
.
so that 
separates 
, 
and 
obtain that 
is equilateral.
and 
obtain that the 
is 
, i.e. 
be an equilateral triangle so that 
separates 
, 
from which results 
and
. Thus, 
. If denote 
, then 
, i.e. 
, where 
and 
. Thus, 
and 
is equilateral triangle because 
and 
. Therefore, 
. Obtain that
is 
is 
, i.e. 
so that 
for which 
. Prove easily that 
is 
, 
and 
is a parallelogram because 
and 
. Therefore, 
is an 
, i.e. 
and 
.
. Observe : that 
is 
; 
is equilateral,
and 
; 
is 
-
; 
is 
. Thus, 
and 
is 
.
. Apply the theorem of Sinus 
![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. Construct the point 
so that 
and the point 
so that
. Observe that 
from where obtain that 
and 
, i.e. 
.
. Apply the Sinus' theorem in the mentioned triangles:
PP2. Let
be a square and consider its interior point 
so that 
. Prove that the triangle 
is an equilateral triangle. Proof 1. Since
obtain that 
. Denote 
and the projection 
of the point 
on the line 
. Prove easily that 
,
, 
and 
. Therefore, the triangle 
is -isosceles. In conclusion, 
![$\frac 12\cdot\left[180^{\circ}-m\left(\widehat{MFB}\right)\right]=$](http://latex.artofproblemsolving.com/b/2/1/b21a1501757b9815237cf7d495f5ae5ba69bf423.png)
, what means the triangle 
is equilateral .Proof 2. Construct the equilateral
, where 
separates and 
. Observe that 
and the ray 
is thebisector of
. Thus, 
. Therefore, 
is equilateral.Proof 3. Denote the projection
of 
on 
.Observe that 
, the quadrilateral 
is inscribed in the circle with the diameter ![$[BD]$](http://latex.artofproblemsolving.com/3/2/6/3261e689901bce018ecdef47b9bc60a78ead3746.png)
and 
.Thus,
is equilateral.Proof 4. Soon.
This post has been edited 81 times. Last edited by Virgil Nicula, Nov 22, 2015, 4:39 PM
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