I really need help on these 2 geometry problems, it would be nice if someone can post solutions:
1. In Triangle ABC, points P and Q lie on sides AB and AC, respectively, such that the circumcircle of Triangle APQ is tangent to side BC at D. Let EE lie on side BCBC such that BD ≅ EC. Line DP intersects the circumcircle of Triangle CDQ again at X, and line DQ intersects the circumcircle of Triangle BDP again at Y. Prove that D,E,X and Y are concyclic.
2. In triangle ABC, points D, E, and F lie on sides BC, CA, and AB, respectively, such that each of the quadrilaterals AFDE, BDEF, and CEFD has an incircle. Prove that the inradius of triangle ABC is twice the inradius of triangle DEF.
Here are some hints:
1. Use bc-inversion.
2. Use power of a point and Sawayama -Thebault.
Is there a solution to problem 1 without inversion? I don't really know inversion, like is there an angle chasing solution? If so can you post a solution?
I've already given you some obvious hints. You should be able to figure out from there.