2004 JBMO Problems/Problem 2
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 .
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: