423. Geometry problems.
by Virgil Nicula, Mar 28, 2015, 11:49 PM
Lemma 1 (well known). Let the triangle 
. Prove that for any point 
there is the relation 
Proof.![$\frac {DB}{DC}=\frac {[ABD]}{[ACD]}=\frac {AB\cdot AD\cdot\sin\widehat{DAB}}{AC\cdot AD\cdot\sin\widehat{DAC}}\implies$](//latex.artofproblemsolving.com/7/8/6/786603616d3abc64bf7db117256ad0919a1f84d6.png)
Lemma 2 (well known). Let be a triangle and the points , 
, 
where 
. Prove that 
.
Proof. Denote
. Therefore, 
, i.e. the required relation.
Lemma 3. Let a convex
with 
Prove that for any 
and 
so that 
there is the relation 
Proof. Denote
Apply the lemma 
PP3. Let a convex pentagon
where 
, 
, 
and 
. Prove that 
.
Proof. Let
and 
. Thus, 
Therefore,
from the relation (*). In conclusion, 
i.e. 
From the relation 
get 
i.e. 
PP4 (Cr.Tello). Let
and equilateral 
so that 
. Let: 
, 
; 
, 
. Prove that 
.
Proof. Prove easily that
, where 
. Let 
.
In conclusion,
.
PP5 (R.A). Let
with 
, 
, 
and 
. Prove that 
.
Proof. I"ll use the well known identities
and 
in any . Thus, 
Thus, 
I"ll prove the first required relation 
. Indeed, 
.
PP6 (R.A). Let with 
and 
. Prove that 
Proof. Denote the incircle and observe that 
Remark that the circumradius of 
has
the length
and 
is the circumcircle of 
. In conclusion, from the relations 
obtain that 
and 
.
I"ll prove the first relation
. Indeed, 
.
PP7 (Miguel Ochoa). Let with 
and 
Prove that 
Proof. Apply
Particular case. If the point
is the incenter of and 
then 
PP8 (Ruben Dario). For an interior point
of let its cevian 
, where , and . For an
interior
of let its cevian , where 
, 
and 
. Prove that 
.
Proof. Denote
and apply the upper lemma 
in for the points 
. Apply the Ceva's theorem to the mentioned points and suitable triangles 
PP9 (Ruben Dario). Let be a rectangle where 
and 
.
Prove that
Proof. Denote
and apply the theorem of Cathetus 
. Thus, 
is cyclically 
and from the theorem of Cathetus in 
obtain that 
. Therefore, 
. In conclusion, 
Apply the theorem of Cathetus in
, i.e. 
and the equivalence 
.
Extension. Let a parallelogram with 
and the points 
so that 
Prove that 
Proof (analytic). Let
. There is 
so that ![$\left\{\begin{array}{c}
F(a\lambda +b,c)\\\\
E[(a+b)\lambda ,c\lambda]\\\\
G\left(\frac {a+b}{2-\lambda},\frac c{2-\lambda}\right)\end{array}\right\|$](//latex.artofproblemsolving.com/f/9/e/f9eec9a0dc6f30a5c0349edc96338e3c6e2ae3e7.png)
. Thus, 
Therefore, 
Hence the equations 
and have at least a common root, i.e.
![$\left[\left(a^2+2ab\right)-\left(b^2+c^2\right)\right]^2=-\left(a^2+2ab\right)\cdot\left[3\left(b^2+c^2\right)-2\left(a^2+2ab\right)\right]$](//latex.artofproblemsolving.com/9/f/b/9fbfb91d66cee547456ca126de2acdf6dfb22e93.png)
.
Denote
. Thus, the previous relation becomes 
Observe that
.
PP10 (Miguel Ochoa Sanchez). Let be a square with the points 
and 
. Prove that 
Proof. Denote
and 
, where 
Thus,
is cyclically 
Remark.
is cyclic 
and 
is cyclic, where 
An easy extension. Let be a rectangle with 
and 
Prove that 
Proof. Denote
, where Thus, is cyclically
PP11 (Ruben Dario). Let the circle
with 
so that 
and 
doesn't separate 
and 
.
Denote
, 
and 
so that 
. Prove that 
Proof 1 (Juan Daniel Valdivia Fuentes). Denote
and observe that 
Apply the Menelaus' theorem to the transversal 
over
the division 
is harmonically. Hence 
Therefore,
and 
More exactly, 
Proof 2 (Ruben Dario). Nice proof ! Denote the midpoints
, of ![$[MA]$](//latex.artofproblemsolving.com/d/d/f/ddf91ec100609c83be5c154de4e74c7649687ba6.png)
, ![$[MD]$](//latex.artofproblemsolving.com/9/7/8/97872638fc3f0f502d3ea893f138d450be76d57e.png)
respectively, i.e. 
and 
. Therefore,
With other words. Let
Thus, 
PP12 (Ruben Dario). Let with incircle 
Denote 
. Prove that 
Proof 1.
Proof 2. Denote
and apply the theorem of Sines in the mentioned triangles
In conclusion, 
PP13 (Miguel Ochoa Sanchez). Let with the incircle Let 
and 
Prove that 
Proof. Let
. Thus, 
and 
Apply the Pyhagoras' theorem to
Hence the relations
The Ptolemy's theorem for 
, i.e. 
Remark. Denote
Define 
Thus, 
![$\frac {uv}w=\frac {u+v+w}3\ge \sqrt[3]{uvw}$](//latex.artofproblemsolving.com/4/b/f/4bfa7e8c85ad4e7cf8852eef76c141b203f579e4.png)
See here the Leo Giugiuc's proof.
PP14 (Ruben Auqui). Let an equilateral
and 
so that 
. Prove that 
.
Proof 1.
, 
, 
the generalized Pythagoras' theorem 
,
i.e.
. Let 
and apply 
for any to 
of 
.
Proof 2. Let
. So and 
. I"ll use the well-known relation (get it easily with Stewart's) 
, i.e. 
. Let 
. Thus, 
. So 
and
.
Proof 3. Let . I"ll use the well-known relation , i.e.
, i.e. 
is the midpoint of ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. So 
.
PP15 (Miguel Ochoa Sanchez). Let be a square what is inscribed in the circle 
. Prove that 
.
Proof 1. Suppose w.l.o.g. that belongs to the small arc ![$\overarc[]{BC}$](//latex.artofproblemsolving.com/a/1/6/a1630e6627fd1370ac41004e3844e8c72c17e345.png)
. Let 
and 
. Thus, 
and from the right
and 
obtain that 
, i.e. 
and 
. Thus, 
. So ![$\sum a^4=\left(\sum a^2\right)^2-2\left[a^2c^2+b^2d^2+\left(a^2+c^2\right)\left(b^2+d^2\right)\right]=$](//latex.artofproblemsolving.com/8/1/b/81b28a82d942d7b0e638f4cecefdfaeb3fde86c5.png)
.
Proof 2. Let
, where 
. Prove easily 
, where 
. So, 
![$\cdot \left[4-\left(\sin^22x+\cos^22x\right)\right]=\frac 32\implies$](//latex.artofproblemsolving.com/2/5/8/2581f0e6cf408b23c22fb461d52fea0c34b128e0.png)
.
PP16 (Miguel Ochoa Sanchez). Let with incircle such that 
and 
. Prove that 
, where is length of circumradius
Proof. I"ll use the well-known identity (can prove it easily !)
, where 
. Thus, 
and 
. In conclusion,
is a 
-right triangle 
.
Remark. Let with the incircle , the 
-excircle and 
. Is well-known that 
, i.e. 
and
(from the power of w.r.t. the circumcircle). Thus, 
.
PP17 (Miguel Ochoa Sanchez). Let and a point which belongs to the small arc ![$\overarc[]{AC}$](//latex.artofproblemsolving.com/3/2/1/321d1e83d833ec725a63feb78320f1930d4eb295.png)
. Prove that 
, where 
is the area of .
Proof. Apply the Ptolemy's theorem
and the generalized Pytagoras' theorem 
. Prove easily that the relations
![$\sin 2B\left[(ax+cz)^2-2acxz\right]\ \stackrel{(*)}{=}$](//latex.artofproblemsolving.com/e/5/8/e585f13e19fbac6ac66d3f604de43d0d55985b30.png)
.
Remark.
.
PP18 (Miguel Ochoa Sanchez). Let with the -excircle and 
. Prove that 
.
Proof.Is well-known that
. Apply the Ceva's theorem to the point and 
. Apply the Menelaus' theorem to the transversals 
.
PP19 (Gustavo Jimmy Garcia Paytan). Let with the incircle 
and the points 
so that the lines 
are tangent to . Denote
the lengths
of the inradii for the triangles 
, 
and 
respectively. Prove that 
.
Proof. Observe that
and 
.
Observe that
. In conclusion, the required relation is truly.
PP20 (Miguel O. Sanchez). Let an-isosceles and its interior so that 
. Let 
. Prove that the ray 
is the -symmedian in 
.
Proof 1 (trigonometric). Denote
and 
. Apply the trigonometric form
of the Ceva's theorem
. Thus, 
.
Proof 2 (trigonometric). Denote
and 
. Apply the Ceva's theorem
to the point and 
. Observe that 
. I"ll use an well-known the relations
.
Proof 3 (metric). Prove easily that
and 
are tangent to the circumcircle of . Denote 
. Hence 
.
Therefore,
.
PP21 (Ruben Dario). Let an-isosceles 
with the incircle which touches given triangle at , and
. Prove that for any 
and 
so that 
is tangent to there are the relations 
.
Proof. Suppose w.l.o.g.
and denote 
, where
. Prove easily that 
. Thus
, what is truly.
![$\frac {\sin \phi}{\cos 2\phi}\cdot\left[\frac {\sin(2\phi -u)}{\cos (\phi -u)}+\frac{\sin(2\phi -v)}{\cos (\phi -v)}\right]=$](//latex.artofproblemsolving.com/8/e/c/8ecad6fec2973a3dfb22ac89390142ba48e81581.png)
![$\frac {\sin \phi}{\cos 2\phi}\cdot\frac {2\sin\phi [1+\cos (u-v)]}{1+\cos (u-v)}=$](//latex.artofproblemsolving.com/2/d/3/2d307dddc35fb0c46b574074d79af2da8e710341.png)
. Prove that for any point 
there is the relation 
Proof.
![$\frac {DB}{DC}=\frac {[ABD]}{[ACD]}=\frac {AB\cdot AD\cdot\sin\widehat{DAB}}{AC\cdot AD\cdot\sin\widehat{DAC}}\implies$](http://latex.artofproblemsolving.com/7/8/6/786603616d3abc64bf7db117256ad0919a1f84d6.png)
Lemma 2 (well known). Let
be a triangle and the points , 
, 
where 
. Prove that 
.Proof. Denote
. Therefore, 
, i.e. the required relation.Lemma 3. Let a convex
with 
Prove that for any 
and 
so that 
there is the relation 
Proof. Denote
Apply the lemma 
PP3. Let a convex pentagon
where 
, 
, 
and 
. Prove that 
.Proof. Let
and 
. Thus, 
Therefore,
from the relation (*). In conclusion, 
i.e. 
From the relation 
get 
i.e. 
PP4 (Cr.Tello). Let
and equilateral 
so that 
. Let: 
, 
; 
, 
. Prove that 
.Proof. Prove easily that
, where 
. Let 
.In conclusion,
.PP5 (R.A). Let
with 
, 
, 
and 
. Prove that 
.Proof. I"ll use the well known identities
and 
in any . Thus, 
Thus, 
I"ll prove the first required relation 
. Indeed, 
.PP6 (R.A). Let
with 
and 
. Prove that 
Proof. Denote the incircle
and observe that 
Remark that the circumradius of 
hasthe length
and 
is the circumcircle of 
. In conclusion, from the relations 
obtain that 
and 
.I"ll prove the first relation
. Indeed, 
.PP7 (Miguel Ochoa). Let
with 
and 
Prove that 
Proof. Apply
Particular case. If the point
is the incenter of and 
then 
PP8 (Ruben Dario). For an interior point
of let its cevian 
, where , and . For aninterior
of let its cevian , where 
, 
and 
. Prove that 
. Proof. Denote
and apply the upper lemma 
in for the points 
. Apply the Ceva's theorem to the mentioned points and suitable triangles 
PP9 (Ruben Dario). Let
be a rectangle where 
and 
.Prove that
Proof. Denote
and apply the theorem of Cathetus 
. Thus, 
is cyclically 
and from the theorem of Cathetus in 
obtain that 
. Therefore, 
. In conclusion, 
Apply the theorem of Cathetus in
, i.e. 
and the equivalence 
.Extension. Let a parallelogram
with 
and the points 
so that 
Prove that 
Proof (analytic). Let
. There is 
so that ![$\left\{\begin{array}{c}
F(a\lambda +b,c)\\\\
E[(a+b)\lambda ,c\lambda]\\\\
G\left(\frac {a+b}{2-\lambda},\frac c{2-\lambda}\right)\end{array}\right\|$](http://latex.artofproblemsolving.com/f/9/e/f9eec9a0dc6f30a5c0349edc96338e3c6e2ae3e7.png)
. Thus, 
Therefore, 
Hence the equations 
and have at least a common root, i.e.
![$\left[\left(a^2+2ab\right)-\left(b^2+c^2\right)\right]^2=-\left(a^2+2ab\right)\cdot\left[3\left(b^2+c^2\right)-2\left(a^2+2ab\right)\right]$](http://latex.artofproblemsolving.com/9/f/b/9fbfb91d66cee547456ca126de2acdf6dfb22e93.png)
.Denote
. Thus, the previous relation becomes 
Observe that
.PP10 (Miguel Ochoa Sanchez). Let
be a square with the points 
and 
. Prove that 
Proof. Denote
and 
, where 
Thus, 
is cyclically 
Remark.
is cyclic 
and 
is cyclic, where 
An easy extension. Let
be a rectangle with 
and 
Prove that 
Proof. Denote
, where Thus, is cyclically 
PP11 (Ruben Dario). Let the circle
with 
so that 
and 
doesn't separate 
and 
.Denote
, 
and 
so that 
. Prove that 
Proof 1 (Juan Daniel Valdivia Fuentes). Denote
and observe that 
Apply the Menelaus' theorem to the transversal 
over
the division 
is harmonically. Hence 
Therefore,
and 
More exactly, 
Proof 2 (Ruben Dario). Nice proof ! Denote the midpoints
, of ![$[MA]$](http://latex.artofproblemsolving.com/d/d/f/ddf91ec100609c83be5c154de4e74c7649687ba6.png)
, ![$[MD]$](http://latex.artofproblemsolving.com/9/7/8/97872638fc3f0f502d3ea893f138d450be76d57e.png)
respectively, i.e. 
and 
. Therefore,
With other words. Let
Thus, 
PP12 (Ruben Dario). Let
with incircle 
Denote 
. Prove that 
Proof 1.
Proof 2. Denote
and apply the theorem of Sines in the mentioned triangles 
In conclusion, 
PP13 (Miguel Ochoa Sanchez). Let
with the incircle Let 
and 
Prove that 
Proof. Let
. Thus, 
and 
Apply the Pyhagoras' theorem to 
Hence the relations
The Ptolemy's theorem for 
, i.e. 
Remark. Denote
Define 
Thus, 
![$\frac {uv}w=\frac {u+v+w}3\ge \sqrt[3]{uvw}$](http://latex.artofproblemsolving.com/4/b/f/4bfa7e8c85ad4e7cf8852eef76c141b203f579e4.png)
See here the Leo Giugiuc's proof.
PP14 (Ruben Auqui). Let an equilateral
and 
so that 
. Prove that 
.Proof 1.
, 
, 
the generalized Pythagoras' theorem 
,i.e.
. Let 
and apply 
for any to 
of 
.Proof 2. Let
. So and 
. I"ll use the well-known relation (get it easily with Stewart's) 
, i.e. 
. Let 
. Thus, 
. So 
and
.Proof 3. Let
. I"ll use the well-known relation , i.e. 
, i.e. 
is the midpoint of ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
. So 
. PP15 (Miguel Ochoa Sanchez). Let
be a square what is inscribed in the circle 
. Prove that 
.Proof 1. Suppose w.l.o.g. that
belongs to the small arc ![$\overarc[]{BC}$](http://latex.artofproblemsolving.com/a/1/6/a1630e6627fd1370ac41004e3844e8c72c17e345.png)
. Let 
and 
. Thus, 
and from the right
and 
obtain that 
, i.e. 
and 
. Thus, 
. So ![$\sum a^4=\left(\sum a^2\right)^2-2\left[a^2c^2+b^2d^2+\left(a^2+c^2\right)\left(b^2+d^2\right)\right]=$](http://latex.artofproblemsolving.com/8/1/b/81b28a82d942d7b0e638f4cecefdfaeb3fde86c5.png)
.Proof 2. Let
, where 
. Prove easily 
, where 
. So, 
![$\cdot \left[4-\left(\sin^22x+\cos^22x\right)\right]=\frac 32\implies$](http://latex.artofproblemsolving.com/2/5/8/2581f0e6cf408b23c22fb461d52fea0c34b128e0.png)
.PP16 (Miguel Ochoa Sanchez). Let
with incircle such that 
and 
. Prove that 
, where is length of circumradiusProof. I"ll use the well-known identity (can prove it easily !)
, where 
. Thus, 
and 
. In conclusion,
is a 
-right triangle 
.
Remark. Let with the incircle , the 
-excircle and 
. Is well-known that 
, i.e. 
and
(from the power of w.r.t. the circumcircle). Thus, 
. PP17 (Miguel Ochoa Sanchez). Let
and a point which belongs to the small arc ![$\overarc[]{AC}$](http://latex.artofproblemsolving.com/3/2/1/321d1e83d833ec725a63feb78320f1930d4eb295.png)
. Prove that 
, where 
is the area of . Proof. Apply the Ptolemy's theorem
and the generalized Pytagoras' theorem 
. Prove easily that the relations 
![$\sin 2B\left[(ax+cz)^2-2acxz\right]\ \stackrel{(*)}{=}$](http://latex.artofproblemsolving.com/e/5/8/e585f13e19fbac6ac66d3f604de43d0d55985b30.png)
.Remark.
.PP18 (Miguel Ochoa Sanchez). Let
with the -excircle and 
. Prove that 
. Proof.Is well-known that
. Apply the Ceva's theorem to the point and 
. Apply the Menelaus' theorem to the transversals 
. PP19 (Gustavo Jimmy Garcia Paytan). Let
with the incircle 
and the points 
so that the lines 
are tangent to . Denotethe lengths
of the inradii for the triangles 
, 
and 
respectively. Prove that 
.Proof. Observe that
and 
.Observe that
. In conclusion, the required relation is truly. PP20 (Miguel O. Sanchez). Let an
-isosceles and its interior so that 
. Let 
. Prove that the ray 
is the -symmedian in 
. Proof 1 (trigonometric). Denote
and 
. Apply the trigonometric formof the Ceva's theorem
. Thus, 
.Proof 2 (trigonometric). Denote
and 
. Apply the Ceva's theoremto the point
and 
. Observe that 
. I"ll use an well-known the relations
.Proof 3 (metric). Prove easily that
and 
are tangent to the circumcircle of . Denote 
. Hence 
.Therefore,
.PP21 (Ruben Dario). Let an
-isosceles 
with the incircle which touches given triangle at , and . Prove that for any 
and 
so that 
is tangent to there are the relations 
.Proof. Suppose w.l.o.g.
and denote 
, where
. Prove easily that 
. Thus 
, what is truly.
![$\frac {\sin \phi}{\cos 2\phi}\cdot\left[\frac {\sin(2\phi -u)}{\cos (\phi -u)}+\frac{\sin(2\phi -v)}{\cos (\phi -v)}\right]=$](http://latex.artofproblemsolving.com/8/e/c/8ecad6fec2973a3dfb22ac89390142ba48e81581.png)
![$\frac {\sin \phi}{\cos 2\phi}\cdot\frac {2\sin\phi [1+\cos (u-v)]}{1+\cos (u-v)}=$](http://latex.artofproblemsolving.com/2/d/3/2d307dddc35fb0c46b574074d79af2da8e710341.png)
This post has been edited 437 times. Last edited by Virgil Nicula, Feb 11, 2018, 3:01 PM
August 2018
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