2018 JBMO Problems/Problem 4
Problem
Let and ,, the symmetrics of vertex over opposite sides.The intersection of the circumcircles of and is . and are defined similarly.Prove that lines , and are concurent.
Solution
passes through circumcenter of .
Since length of is equal to that of , it follows that circumcenter of lies on side of .
Similarly, circumcenter of lies on side of .
So the radical axis () of circles and passes through . Also note that Mid-point of and center of circle ACC' () passes through and is perpendicular to .
Similarly, radical axis () of circles and passes through . Also, Mid-point of and center of circle ABB' () passes through and is perpendicular to .
Thus, circumcenter, of is the orthocenter of .
Thus, is perpendicular to . Thus, passes through , implying passes through circumcenter of .
Now using above, it can be proven similarly that and also pass through circumcenter of .
Thus, lines , and are concurrent at circumcenter of .