IMO 2018 1first 's problem solution

Let acute-angled triangle , and let   be the circumscribed circle. Points and in the parts and corresponding order .

The perpendicular bisector of and intersecting the small arches and the in points and respectively.

Prove that the lines and are either parallel or are identical.

Solution

Suppose that intersects the circumscribed circle of a point .

We observe that .

Thus the triangle is isosceles with .

Similarly if intersects circumcircle of at , then .

We will prove that , ie .

We note that , while (  is the core of the circle with center and radius .

So it suffices to prove that , that .

But we know that , by definition .

So long .

We observe that the quadrilateral is recordable and therefore the is the bisector .

If it was not equilateral, then since it belongs to the bisector and , should it be writable (south pole theorem) inappropriate, since the circumscribed circle is the circumscribed circle and does not belong to this circle.

So it is isosceles with and the requested follow.