1.) Explain how to construct a square with the same area of any given quadrilaterals.
2.) Let be a point in . Extend intersect at point respectively. If and are cyclic quadrilaterals, prove that is also cyclic quadrilateral.
3.) Use Ceva's theorem to prove that the 3 angle bisectors of intersect at 1 point.
4.) Let with point as incenter. Extend intersect circumcircle of at point .
4.1) If point is the center of excircle of which is opposite to angle , prove that is the center of the circumcenter of
4.2) Prove that
5.) Let has an incircle, which is tangent to at point respectively. Extend intersect at point . Prove that point are collinear.
This is the part of Thailand 1st round math POSN problems.