In this note, I have a selection of questions for you to prove. It is for all levels to try. The theme is geometry. Good luck.
1) Scalene triangle is right-angled at . The tangent to the circumcircle of triangle at point intersects at . Let the points of contact of the incircle of triangle with sides and be and respectively. Let and at and at .
Prove that triangle is both obtuse and isosceles.
2) Let be a square with centre . Let be a square and let be the square containing in its interior.
Prove that is the midpoint of .
3) In triangle it is known that and . The bisector of angle intersects at . Let be the midpoint of . Let be the foot of the perpendicular from to . The perpendicular bisector of intersects at .
Prove that .
4) Point lies outside circle . A line through intersects at points and . Points and are the points of contact of the two tangents from to . The line passing through and parallel to intersects at .
Prove that intersects the segment at its midpoint.
5) Acute triangle has circumcircle . The tangent at to intersects at . Let be the midpoint of the segment . Let be the second intersection point of with . Let be the second intersection point of with .
Prove that is parallel to .
6) Prove that if 3 congruent circles pass through the same point, then their other three intersection points lie on a fourth circle with the same radius.