Write a full solution.
Let be orthocenter of and point be a midpoint between each vertex and respectively. If is a point on circumcircle of not on the vertex , and be the midpoint between . Prove that lie on the same circle.
Let the tangents of circumcircle of at point intersect at point . Prove that is symmedian line of .
Let be 2 points that are isogonal conjugate to each other in . If and is perpendicular to at point and respectively, and is perpendicular to at point and respectively, and and is perpendicular to at point and respectively. Prove that lie on the same circle, and the center of that circle lies in the midpoint of and .
Let be the center of incircle, nine-point circle, and orthocenter of respectively. Construct perpendicular to at point respectively. If intersect circumcircle of at point such that is a midpoint of . Prove that where is an inradius of .
Let be a foot of altitude of from point . If are reflection of by respectively, such that intersect at point respectively. Prove that is perpendicular to respectively.
This note is a part of Thailand Math POSN 2nd round 2015.