# 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 11:02, 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,