433. Some properties of remarkable lines in a triangle II.
by Virgil Nicula, Nov 16, 2015, 2:27 PM
P1. Let 
, the midpoint 
of ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, 
and intersection 
of tangent lines at 
, 
to circle with diameter ![$[AM]$](//latex.artofproblemsolving.com/1/f/9/1f9b22599237fb6240a50b5f75e8f6ced1292374.png)
. Prove that 
.
Proof 1 (synthetical). Let
be a point for which 
and 
. Denote 
. Observe that ![$[MAB]=[MAC]$](//latex.artofproblemsolving.com/8/9/d/89d2236039a5f13859756da620347cce793b03c1.png)
and![$\left\{\begin{array}{c}
[ABL]=[AML]=[ACL]\\\\
\widehat{AEL}\equiv\widehat{AML}\equiv\widehat{AFL}\end{array}\right\|$](//latex.artofproblemsolving.com/4/9/3/4932939b52b69a1698ba2a8fae0f50bdf778003e.png)
, i.e. 
is an harmonic cyclic quadrilateral.
From an well-known property obtain that
- the bisector of the side , i.e. 
. I was inspired by phuongtheong. Thank you.
Proof 2. Let
, the circle 
with the diameter and the intersections 
, 
of the perpendicular bisector of the side with the tangent lines to t at ,
respectively. We observe that
and 
, i.e. 
. From the Sinus' theorem applied in 
, 
results 
and
. Therefore, the point belongs to the perpendicular bisector of the side 
, what is truly, from the relation 
and the identity posted here
Remark. Prove easily that
, 
. Since 
obtain that
.
Otherwise.
, where 
is the intersection of the tangent lines at 
, 
to the circumcirle of .
P2. Let
be an 
-isosceles triangle. For 
denote second intersection 
of the line 
with the circumcircle of . Prove that 
, 
.
Proof 1.
, 
.
Proof 2 (trigonometric). Denote
. Observe that 
, 
and 
. Thus, 
, what is truly.
Extension. Let and for denote the second intersection of the line with the circumcircle of . Prove that 
.
Proof.
.
P3. Let and the midpoints , 
of ![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
, ![$[AC]$](//latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
respectively and the intersections , 
of the -symmedian with 
, 
respectively. Prove that 
.
Proof 1 (metric). Denote
. It's well-known that 
. Apply the Menelaus' theorem to the transversals:
is cyclic 
.
Proof 2 (metric). Denote . Apply the Ceva's theorem to the points and 
is cyclic .
Remark. Denote
. Apply an well-known relation 
.
P4. Let , the midpoint of , 
which satisfy 
. The bisector of 
meet 
at . Let 
. Prove that 
.
Proof 1 (metric). Observe that
and 
. 
. Thus, 
.
Apply the Menelaus' theorem to the transversal
and 
.
Proof 2 (synthetic). Prove easily that that
. Denote 
for which 
. Since 
and 
obtain that is the midpoint of ![$[ED]$](//latex.artofproblemsolving.com/9/2/9/929b45305c082cd1e656a5a9897f0df75740e43e.png)
, i.e.
. Notice that the division 
is harmonically 
, where 
is an harmonical division. Since
is the midpoint of
obtain 
, i.e. 
. Hence 
. Together with 
this means that 
is an isosceles triangle and
. In conclusion, 
.
Lemma. Let with incenter 
and 
, 
so that 
. Then 
.
Denote
, the line 
so that 
and 
, 
. Suppose w.l.o.g. 
and 
. Thus, 
.[/hide]
P5. Let
be a bisector of , where 
. The line trough the incenters of 
and 
meets in and 
in . Prove that 
.
Proof. Is well-known that
. Denote 
and apply the upper lemma: 
. Since 
obtain that 
.
P6. Let and , , be the tangent points of the incircle 
with , 
, respectively. meets again at . Show that 
bisects 
.
Proof 1. Let
.
Proof 2 (metric).
. Prove easily that 
. In conclusion, 
.
Lemma. Let and an interior for which denote 
. Prove that for any
and 
so that 
exists the relation 
.
Remark. In the proof of PP apply the particular cases:
.
P7. Let an acute with orthocenter 
, incenter and circumcenter 
and 
. Prove that 
Proof.
. 
. In conclusion,
.
P8. Given . The bisector of 
meets and circumcircle of at , respectively. Let , be midpoints of 
, 
respectively. The circumcircle of
meets again 
at . The circle passing through and which is tangent to at meets 
, respectively at , 
. Show that 
belong to same line.
Proof. Prove easily that
. Thus, 
.
Therefore,
. Thus, 
. Since 
is cyclically obtain that 
. In conclusion, 
.
P9. Let
be the -bisector of , where . The incircles of the triangles 
, 
touch at the points , respectively. Prove that 
.
Proof. I"ll use the relations
and 
. Thus, 
and 
truly.
P10. Let with 
and 
so that 
. Prove that 
.
Proof. Denote
and apply an well-known identity 
. Apply the Menelaus' theorem to the transversals:
.
P11 (Samuel Palacios Paulino). Let with the incircle which touches , at , respectively. Denote the midpoint of , the
projection of on , the point 
so that 
and the intersections 
, 
. Prove that 
.
Proof. Suppose w.l.o.g. that
. Observe that 
. The triangles 
and 
have 
and
. From an well-known property obtain that 
Remark 1.
Remark 2. Apply the Menelaus' theorem to
P12 (Samuel Palacios Paulino). Let with the incircle 
which touches , at , respectively
and
, so that 
and 
is the bisector of 
. Prove that 
.
Proof. Prove easily that
. I"ll use the length 
of the -bisector in the standard 
. Therefore,
. With 
, i.e. 
, the relation becomes
Remark. Prove similarly that an easy its extension
, where 
. Very nice problem!
P13. Let with orthic 
where , 
, , orthocenter and ,
so that 
and 
is the bisector of 
. Prove that 
.
Proof. Prove easily that
. I"ll use the length of the -bisector in the standard . Therefore,
.
P14..Let
be the circumcircle of and 
be the diameter of . Consider collinear points 
, 
, 
. Prove that 
Proof. Let
we have 
therefore 
is cyclic hence 
hence 
is cyclic. By similarity
of triangles
and 
we have : 
.Now consider . Hence 
and 
.
Hence
, i.e. is cyclic. Therefore 
. Therefore is cyclic then 
Lemma. For let incenter , -excenter 
reflection 
of w.r.t. the circumcenter and second intersection of 
with the circumcircle.
Incircle and- excircle touch in and 
respectively. Then the intersections 
and 
belong to the circumcircle.
Proof. Denote
- the power of the point 
w.r.t. the circumcircle 
of the triangle 
I recall
the well-known relations :
and 
and 
, i.e. 
and 
, i.e. 
Observe the "identity" between the two above proofs : 
Enunciation. For denote the incenter , the - exincenter and the reflection of the point w.r.t. the circumcenter.
Define the intersections :
. Prove that : 
Proof. I"ll use the notations from the above lemma. Apply the Pascal's theorem in the following two pairs of hexagons :
P15. Let an acute with the orthocenter and the orthic , where , 
and 
. Prove that 
Denote 
. Prove 
.
Denote the points 
so that 
and 
. Prove that 
.
Proof.
Let , be second intersections of 
, 
respectively with the circumcircle of 
. Thus, 
cyclic, i.e. 
, i.e. 
, i.e. 
is cyclic.
.
P16. Let and such that bisects angle . IF 
, 
and 
, THEN find and .
Proof 1.
. Let the common second point 
of with the circumcircle of . Thus, 
Thus,
, i.e. 
. Therefore, 
. In conclusion, 
.
Proof 2. Exists
so that 
. Apply the Stewart's relation 
.
Proof 3. Exists so that 
. Apply the generalized theorem of Cosines 
.
Remark. If
and 
, then 
.
P17. Let with the incenter . The circumcircle of 
cut the sidelines , in the points , respectively. Prove that 
.
Proof 1. Suppose w.l.o.g.
, i.e. 
and 
. Thus, the -excenter 
, ![$[I_aI]$](//latex.artofproblemsolving.com/2/3/0/2300ecd105ea61e6ca96618a1295a7c71c89bbae.png)
is a diameter of and 
Apply the power of point w.r.t. 
Proof 2.
, where is diameter of and 
is the bisector of 
, 
and 
.
P18. Let an acute with incircle 
and circumcircle 
. The circles 
and 
are tangent internally to 
in the same point 
.
The circle
is tangent externally to the circle 
and is tangent internally to the circle 
. Prove that 
, where 
is the semiperimeter and ![$ S=[ABC]$](//latex.artofproblemsolving.com/a/c/5/ac5489c655a5d312c5c91c4f77c55eadde984421.png)
is the area for
.
Remark. Prove easily that
. The relations 
are well-known.
Proof 1. Denote
. Apply the Stewart's theorem in 
for the cevian-rays 
and 
.
Proof 2. Apply the Pythagoras' theorem in the triangles
Therefore,
, i.e. 
, where 
is the - exinradius of .
Remark. Prove easily that two more interesting relations :
and 
, 
.
.
, the midpoint 
of ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
, 
and intersection 
of tangent lines at 
, 
to circle with diameter ![$[AM]$](http://latex.artofproblemsolving.com/1/f/9/1f9b22599237fb6240a50b5f75e8f6ced1292374.png)
. Prove that 
.Proof 1 (synthetical). Let
be a point for which 
and 
. Denote 
. Observe that ![$[MAB]=[MAC]$](http://latex.artofproblemsolving.com/8/9/d/89d2236039a5f13859756da620347cce793b03c1.png)
and
![$\left\{\begin{array}{c}
[ABL]=[AML]=[ACL]\\\\
\widehat{AEL}\equiv\widehat{AML}\equiv\widehat{AFL}\end{array}\right\|$](http://latex.artofproblemsolving.com/4/9/3/4932939b52b69a1698ba2a8fae0f50bdf778003e.png)
, i.e. 
is an harmonic cyclic quadrilateral.From an well-known property obtain that
- the bisector of the side , i.e. 
. I was inspired by phuongtheong. Thank you.Proof 2. Let
, the circle 
with the diameter and the intersections 
, 
of the perpendicular bisector of the side with the tangent lines to t at , respectively. We observe that
and 
, i.e. 
. From the Sinus' theorem applied in 
, 
results 
and
. Therefore, the point belongs to the perpendicular bisector of the side 
, what is truly, from the relation 
and the identity posted hereRemark. Prove easily that
, 
. Since 
obtain that
.Otherwise.
, where 
is the intersection of the tangent lines at 
, 
to the circumcirle of .P2. Let
be an 
-isosceles triangle. For 
denote second intersection 
of the line 
with the circumcircle of . Prove that 
, 
.Proof 1.
, 
.Proof 2 (trigonometric). Denote
. Observe that 
, 
and 
. Thus, 
, what is truly.Extension. Let
and for denote the second intersection of the line with the circumcircle of . Prove that 
.Proof.
.P3. Let
and the midpoints , 
of ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
, ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
respectively and the intersections , 
of the -symmedian with 
, 
respectively. Prove that 
.Proof 1 (metric). Denote
. It's well-known that 
. Apply the Menelaus' theorem to the transversals:
is cyclic 
.Proof 2 (metric). Denote
. Apply the Ceva's theorem to the points and 
is cyclic .Remark. Denote
. Apply an well-known relation 
.P4. Let
, the midpoint of , 
which satisfy 
. The bisector of 
meet 
at . Let 
. Prove that 
.Proof 1 (metric). Observe that
and 
. 
. Thus, 
.Apply the Menelaus' theorem to the transversal
and 
.Proof 2 (synthetic). Prove easily that that
. Denote 
for which 
. Since 
and 
obtain that is the midpoint of ![$[ED]$](http://latex.artofproblemsolving.com/9/2/9/929b45305c082cd1e656a5a9897f0df75740e43e.png)
, i.e.
. Notice that the division 
is harmonically 
, where 
is an harmonical division. Since is the midpoint of
obtain 
, i.e. 
. Hence 
. Together with 
this means that 
is an isosceles triangle and
. In conclusion, 
.Lemma. Let
with incenter 
and 
, 
so that 
. Then 
.Denote
, the line 
so that 
and 
, 
. Suppose w.l.o.g. 
and 
. Thus, 
.[/hide]P5. Let
be a bisector of , where 
. The line trough the incenters of 
and 
meets in and 
in . Prove that 
.Proof. Is well-known that
. Denote 
and apply the upper lemma: 
. Since 
obtain that 
.P6. Let
and , , be the tangent points of the incircle 
with , 
, respectively. meets again at . Show that 
bisects 
.Proof 1. Let
.Proof 2 (metric).
. Prove easily that 
. In conclusion, 
.Lemma. Let
and an interior for which denote 
. Prove that for any
and 
so that 
exists the relation 
.Remark. In the proof of PP apply the particular cases:
.P7. Let an acute
with orthocenter 
, incenter and circumcenter 
and 
. Prove that 
Proof.
. 
. In conclusion,
.P8. Given
. The bisector of 
meets and circumcircle of at , respectively. Let , be midpoints of 
, 
respectively. The circumcircle of meets again 
at . The circle passing through and which is tangent to at meets 
, respectively at , 
. Show that 
belong to same line.Proof. Prove easily that
. Thus, 
.Therefore,
. Thus, 
. Since 
is cyclically obtain that 
. In conclusion, 
.P9. Let
be the -bisector of , where . The incircles of the triangles 
, 
touch
at the points , respectively. Prove that 
.Proof. I"ll use the relations
and 
. Thus, 
and 
truly.P10. Let
with 
and 
so that 
. Prove that 
.Proof. Denote
and apply an well-known identity 
. Apply the Menelaus' theorem to the transversals:
.P11 (Samuel Palacios Paulino). Let
with the incircle which touches , at , respectively. Denote the midpoint of , theprojection
of on , the point 
so that 
and the intersections 
, 
. Prove that 
. Proof. Suppose w.l.o.g. that
. Observe that 
. The triangles 
and 
have 
and
. From an well-known property obtain that 
Remark 1.
Remark 2. Apply the Menelaus' theorem to
P12 (Samuel Palacios Paulino). Let
with the incircle 
which touches , at , respectivelyand
, so that 
and 
is the bisector of 
. Prove that 
. Proof. Prove easily that
. I"ll use the length 
of the -bisector in the standard 
. Therefore,
. With 
, i.e. 
, the relation becomes
Remark. Prove similarly that an easy its extension
, where 
. Very nice problem! P13. Let
with orthic 
where , 
, , orthocenter and , so that 
and 
is the bisector of 
. Prove that 
. Proof. Prove easily that
. I"ll use the length of the -bisector in the standard . Therefore,
.P14..Let
be the circumcircle of and 
be the diameter of . Consider collinear points 
, 
, 
. Prove that 
Proof. Let
we have 
therefore 
is cyclic hence 
hence 
is cyclic. By similarityof triangles
and 
we have : 
.Now consider . Hence 
and 
.Hence
, i.e. is cyclic. Therefore 
. Therefore is cyclic then 
Lemma. For
let incenter , -excenter 
reflection 
of w.r.t. the circumcenter and second intersection of 
with the circumcircle.Incircle and
- excircle touch in and 
respectively. Then the intersections 
and 
belong to the circumcircle.Proof. Denote
- the power of the point 
w.r.t. the circumcircle 
of the triangle 
I recallthe well-known relations :
and 
and 
, i.e. 
and 
, i.e. 
Observe the "identity" between the two above proofs : 
Enunciation. For
denote the incenter , the - exincenter and the reflection of the point w.r.t. the circumcenter.Define the intersections :
. Prove that : 
Proof. I"ll use the notations from the above lemma. Apply the Pascal's theorem in the following two pairs of hexagons :
P15. Let an acute
with the orthocenter and the orthic , where , 
and 
. Prove that 
Denote 
. Prove 
.
Denote the points 
so that 
and 
. Prove that 
.Proof.
Let , be second intersections of 
, 
respectively with the circumcircle of 
. Thus, 
cyclic, i.e. 
, i.e. 
, i.e. 
is cyclic. 
.P16. Let
and such that bisects angle . IF 
, 
and 
, THEN find and .Proof 1.
. Let the common second point 
of with the circumcircle of . Thus, 
Thus,
, i.e. 
. Therefore, 
. In conclusion, 
.Proof 2. Exists
so that 
. Apply the Stewart's relation 
.Proof 3. Exists
so that 
. Apply the generalized theorem of Cosines 
.Remark. If
and 
, then 
.P17. Let
with the incenter . The circumcircle of 
cut the sidelines , in the points , respectively. Prove that 
. Proof 1. Suppose w.l.o.g.
, i.e. 
and 
. Thus, the -excenter 
, ![$[I_aI]$](http://latex.artofproblemsolving.com/2/3/0/2300ecd105ea61e6ca96618a1295a7c71c89bbae.png)
is a diameter of and 
Apply the power of point w.r.t. 
Proof 2.
, where is diameter of and 
is the bisector of 
, 
and 
.P18. Let an acute
with incircle 
and circumcircle 
. The circles 
and 
are tangent internally to 
in the same point 
.The circle
is tangent externally to the circle 
and is tangent internally to the circle 
. Prove that 
, where 
is the semiperimeter and ![$ S=[ABC]$](http://latex.artofproblemsolving.com/a/c/5/ac5489c655a5d312c5c91c4f77c55eadde984421.png)
is the area for
.Remark. Prove easily that
. The relations 
are well-known.Proof 1. Denote
. Apply the Stewart's theorem in 
for the cevian-rays 
and 
.Proof 2. Apply the Pythagoras' theorem in the triangles
Therefore,
, i.e. 
, where 
is the - exinradius of .Remark. Prove easily that two more interesting relations :
and 
, 
..
This post has been edited 118 times. Last edited by Virgil Nicula, Aug 12, 2017, 8:57 AM
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