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be a triangle such that 
. Points 
lie on segments 
, respectively, such that lines 
and 
are the angle bisectors of triangle 
. Find the measure of angle 
.
satisfy 
prove in equality.
has circumcircle 
and circumcenter 
. A circle 
with center 
intersects the segment 
at points 
and 
, such that 
, , , and 
are all different and lie on line in this order. Let 
and 
be the points of intersection of and , such that , , , , and lie on in this order. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
and 
are different and intersect at the point 
. Prove that lies on the line 
.
be a point inside a triangle 
such that![\[ \angle APB - \angle ACB = \angle APC - \angle ABC.
\]](http://latex.artofproblemsolving.com/a/6/e/a6e3dc27d0457682fa22af4f918f58d9cd8403bc.png)
, 
be the incenters of triangles 
, 
, respectively. Show that the lines 
, 
, 
meet at a point.
, let point be the midpoint of the hypotenuse and lie point 
lie on the side such that 
. Let segments 
and 
intersect at point 
. Prove that the line 
touches the circumscribed circle of the triangle 
.
, let point be the midpoint of the hypotenuse and lie point lie on the side such that . Let segments and intersect at point . Prove that the line touches the circumscribed circle of the triangle .
be a perpendicular foot from to .
and 
., then 
.![\[
ME: MC=\frac{1}{2}=KE: AC,
\]](http://latex.artofproblemsolving.com/b/3/6/b360dd47154676fa49527d9fba3af32459b9818d.png)
which implies that 
.![\[
\angle{KME}=\angle{AMC}\implies \angle{AMB}=\angle{KMC}.
\]](http://latex.artofproblemsolving.com/f/2/b/f2b75aeb20d8c41301eb80f532ea0c5b8b89fd5f.png)
Also, since , and lie on the same circle with centre , then 
.![\[
180^\circ-\angle{KCM}-\angle{KMC}=180^\circ-\angle{ABM}-\angle{AMB} \implies \angle{MAB}=\angle{MKC}.
\]](http://latex.artofproblemsolving.com/f/a/a/faa0bf95734ff4c4f6445ec8a5d98229df9d098a.png)
Thus, 
, which means the line touches the circumscribed circle of the triangle .
