157. Steinbart's theorem. Applications.
by Virgil Nicula, Oct 16, 2010, 7:40 AM
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Lemma 1. Let
be a cyclical hexagon (convex or not).
Then
.
Proof 1. Suppose that
. From the product of evident relations 
obtain the required relation
. Reciprocally, suppose that the relation 
is true. Denote 
and 
,
where
is the circumcircle of the given hexagon. Appling the direct implication which was proved above obtain 
.
From the relations and 
results 
. Since 
obtain 
. Since these triangles have same circumcircle
(similarity ratio is equal to
) obtain 
, i.e. 
and 
. In conclusion, 
, i.e. 
.
Proof 2. Suppose w.l.o.g. that the quadrilaterals
and 
are convex. Denote , 
. Using the well-known relations
(in 
) and 
(in 
) obtain that 
. Using the Ptolemy's relation in the cyclical quadrilaterals and obtain the relations
and 
. Therefore, 
.
Remark. Using the previous result for the cyclical quadrilaterals
and 
obtain the following chain of equivalencies :
.
Lemma 2. Let
, 
be the tangent lines from 
to the circle , where 
. Consider the points 
, 
so 
separates , and doesn't separate , . Denote 
. Then exists the relation 
.
Proof. Denote
. Apply the Sinus' theorem in the triangles 
and 
, i.e. 
.
Lemma 3. Let , 
, 
be three points which belong to the outside of the triangle 
. Denote 
.
Then
(it is the trigonometric form of the Ceva's theorem).
Steinbart's theorem. Let be a triangle with the incircle which touches it at 
, 
, 
.
Consider three points
. Then 
.
Proof. Using the lemma 2 obtain
.
Apply the lemma 2 to the cyclical hexagon
, i.e. 
.
Thus the relation becomes 
. Using the last lemma obtain that 
.
Applications.
PP1.The incircle of
touches its side 
at 
. Let 
be the midpoint of the altitude 
of triangle 
. The line 
meets
the incircle of at a point 
(apart from ). Show that the circumcircle of 
is tangent to the incircle of at the point .
Proof. Suppose w.l.o.g.
. Denote the A-exincircle 
, the point 
, the intersection 
between the line with the tangent
to the incircle in the point and 
. Prove easily that 
, 
and 
. Since 
obtain that 
, i.e. 
. The points 
belong to the circle with the diameter 
. Since
obtain that 
. Thus 
, i.e. the line 
is tangent to the circumcircle of the triangle 
.
PP2. Let
be the incircle in the triangle 
. For all 
we make the following constructions (all indices are considered modulo 3):
is the circle tangent to which passes through the points 
and 
; 
is the point of tangency between and ; finally, the common
tangent in of and intersects the line 
in the point 
. Prove that 
and 
.
Proof.
Lemma 1. Let
be a cyclical hexagon (convex or not).Then
. Proof 1. Suppose that
. From the product of evident relations 
obtain the required relation
. Reciprocally, suppose that the relation 
is true. Denote 
and 
,where
is the circumcircle of the given hexagon. Appling the direct implication which was proved above obtain 
.From the relations
and 
results 
. Since 
obtain 
. Since these triangles have same circumcircle (similarity ratio is equal to
) obtain 
, i.e. 
and 
. In conclusion, 
, i.e. 
.Proof 2. Suppose w.l.o.g. that the quadrilaterals
and 
are convex. Denote , 
. Using the well-known relations
(in 
) and 
(in 
) obtain that 
. Using the Ptolemy's relation in the cyclical quadrilaterals and obtain the relations
and 
. Therefore, 
.Remark. Using the previous result for the cyclical quadrilaterals
and 
obtain the following chain of equivalencies : 
.Lemma 2. Let
, 
be the tangent lines from 
to the circle , where 
. Consider the points 
, 
so 
separates
, and doesn't separate , . Denote 
. Then exists the relation 
.Proof. Denote
. Apply the Sinus' theorem in the triangles 
and 
, i.e. 
.Lemma 3. Let
, 
, 
be three points which belong to the outside of the triangle 
. Denote 
.Then
(it is the trigonometric form of the Ceva's theorem).Steinbart's theorem. Let
be a triangle with the incircle which touches it at 
, 
, 
.Consider three points
. Then 
.Proof. Using the lemma 2 obtain
.Apply the lemma 2 to the cyclical hexagon
, i.e. 
.Thus the relation
becomes 
. Using the last lemma obtain that 
.Applications.
PP1.The incircle of
touches its side 
at 
. Let 
be the midpoint of the altitude 
of triangle 
. The line 
meetsthe incircle
of at a point 
(apart from ). Show that the circumcircle of 
is tangent to the incircle of at the point .Proof. Suppose w.l.o.g.
. Denote the A-exincircle 
, the point 
, the intersection 
between the line with the tangentto the incircle in the point
and 
. Prove easily that 
, 
and 
. Since 
obtain that 
, i.e. 
. The points 
belong to the circle with the diameter 
. Since
obtain that 
. Thus 
, i.e. the line 
is tangent to the circumcircle of the triangle 
.PP2. Let
be the incircle in the triangle 
. For all 
we make the following constructions (all indices are considered modulo 3):
is the circle tangent to which passes through the points 
and 
; 
is the point of tangency between and ; finally, the commontangent in
of and intersects the line 
in the point 
. Prove that 
and 
.Proof.
This post has been edited 50 times. Last edited by Virgil Nicula, Dec 1, 2015, 10:12 AM
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