Difference between revisions of "2004 JBMO Problems/Problem 2"
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− | Let length of side <math>CB | + | Let length of side <math>CB = x</math> and length of <math>QM = a</math>. We shall first prove that <math>QM = QB</math>. |
Let <math>O</math> be the circumcenter of <math>\triangle ACB</math> which must lie on line <math>Z</math> as <math>Z</math> is a perpendicular bisector of isosceles <math>\triangle ACB</math>. | Let <math>O</math> be the circumcenter of <math>\triangle ACB</math> which must lie on line <math>Z</math> as <math>Z</math> is a perpendicular bisector of isosceles <math>\triangle ACB</math>. |
Latest revision as of 12:02, 18 December 2018
Problem
Let be an isosceles triangle with
, let
be the midpoint of its side
, and let
be the line through
perpendicular to
. The circle through the points
,
, and
intersects the line
at the points
and
. Find the radius of the circumcircle of the triangle
in terms of
.
Solution
Let length of side and length of
. We shall first prove that
.
Let be the circumcenter of
which must lie on line
as
is a perpendicular bisector of isosceles
.
So, we have .
Now is a cyclic quadrilateral by definition, so we have:
and,
, thus
, so
.
Therefore in isosceles we have that
.
Let be the circumradius of
.
So we have
or
Now applying Ptolemy's theorem in cyclic quadrilateral , we get:
or,
or,