So a week before I went for the Sharygin Finals, I striked upon this beautiful theorem, called Casey's Theorem (also known as Generalised Ptolemy's Theorem). So here is the theorem, in all its might:

THEOREM (Casey) Given four circles , let denote the length of a common tangent (either internal or external) between and . Then the four circles are tangent to a fifth circle (or line) if and only if for appropriate choice of signs, .

For a proof, see this handout by Luis Gonzales or have a look at Problem 239 and 240 of Problems in Plane Geometry by I. F. Sharygin.

Anyway, I'll just post a few problems which are reduced to mere computations by this theorem (mainly the if part).

PROBLEM 1 (Sharygin Finals 2017 Problem 9.4) Points and are chosen on lateral sides of an isosceles triangle and point is chosen on such that is a parallelogram. Let the lines and meet at point , and let be the intersection points of with the perpendicular line from to . Prove that the circle with center and radius and the circumcircle of triangle are tangent.

SOLUTION

PROBLEM 2 (Tuymaada 2018 Senior/Junior League Problem 8) Quadrilateral with perpendicular diagonals is inscribed in a circle with centre . The tangents to this circle at and together with line form the triangle . Prove that the circumcircles of and are tangent.

SOLUTION

PROBLEM 3 (Source=buratinogigle) Let be a triangle inscribed in circle The circle passes through with is on perpendicular bisector of Similarly, we have the circles and Prove that the circle is tangent to and internally then is tangent to .

SOLUTION

PROBLEM 4 (Source=buratinogigle) Let be a triangle inscribed in circle with altitude . Incircle touches at . is midpoint of . is projection of on . Prove that circle is tangent to .

SOLUTION

PROBLEM 5 (Source=buratinogigle) Let be a triangle with circumcenter and altitude meets at and meets the circle again at is the midpoint of is the projection of on line Prove that the circle is tangent to the circle .

SOLUTION