230. Some usual synthetical properties in a triangle.
by Virgil Nicula, Feb 24, 2011, 1:58 PM
PP1. Let 
with incircle 
. Note 
, 
, 
. Prove that 
.
Proof 1.
. In conclusion, 
.
Proof 2.
is cyclically 
.
PP2. Let with incircle 
. Note midpoint 
of ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
and 
for which 
. Prove that 
.
Proof. Note
and 
. Apply the Menelaus' theorem to the transversal 
.
Remark. Denote
, the diameter ![$[SN]$](//latex.artofproblemsolving.com/4/2/c/42c406d7fff970657b35445127481e58df4f1984.png)
of 
and 
. Prove easily that 
,
i.e.
, where 
is 
-exincircle of . Also 
and 
.
PP3. Let with incircle . Denote midpoint of , 
, 
, . Prove that 
.
Proof 1. Denote
, 
. Observe that 
, 
and 
. Using the well-known relation
obtain that 
. Observe that 
and 
. Using the well-known relation
obtain that 
. From the relations 
and 
obtain that 
, i.e. .
Proof 2. Denote
for which 
, , 
and 
. Observe that 
. Using the well-known relation 
obtain that 
, i.e. 
. From the relations 
and 
obtain that 
, i.e. .
Proof 3. Denote and 
. Observe that , and .
Using the well-known relation obtain that 
. Observe that . Using the well-known
relation
obtain that 
, i.e. 
. In conclusion, .
PP4 (nice). Let with circumcircle 
and incircle . Let , 
, 
, 
. Prove that 
.
Proof. Observe that
and 
because the power 
of 
w.r.t. 
is given by the relation 
.
Therefore,
from where obtain 
, i.e. the point belongs to the circle .
Remark.
.
Prove easily that
and show analogously that 
for which 
.
with incircle 
. Note 
, 
, 
. Prove that 
.Proof 1.
. In conclusion, 
.Proof 2.
is cyclically 
. PP2. Let
with incircle 
. Note midpoint 
of ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
and 
for which 
. Prove that 
.Proof. Note
and 
. Apply the Menelaus' theorem to the transversal 
.Remark. Denote
, the diameter ![$[SN]$](http://latex.artofproblemsolving.com/4/2/c/42c406d7fff970657b35445127481e58df4f1984.png)
of 
and 
. Prove easily that 
,i.e.
, where 
is 
-exincircle of . Also 
and 
.PP3. Let
with incircle . Denote midpoint of , 
, 
, . Prove that 
.Proof 1. Denote
, 
. Observe that 
, 
and 
. Using the well-known relation
obtain that 
. Observe that 
and 
. Using the well-known relation
obtain that 
. From the relations 
and 
obtain that 
, i.e. .Proof 2. Denote
for which 
, , 
and 
. Observe that 
. Using the well-known relation 
obtain that 
, i.e. 
. From the relations 
and 
obtain that 
, i.e. .Proof 3. Denote
and 
. Observe that , and .Using the well-known relation
obtain that 
. Observe that . Using the well-knownrelation
obtain that 
, i.e. 
. In conclusion, .PP4 (nice). Let
with circumcircle 
and incircle . Let , 
, 
, 
. Prove that 
.Proof. Observe that
and 
because the power 
of 
w.r.t. 
is given by the relation 
.Therefore,
from where obtain 
, i.e. the point belongs to the circle .Remark.
.Prove easily that
and show analogously that 
for which 
. This post has been edited 47 times. Last edited by Virgil Nicula, Nov 22, 2015, 2:43 PM
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