Given triangle , its circumcircle , orthorcenter and an arbitrary point $latex P$ . intersect again at , respectively. Let be the reflections of across the lines . Prove that are concyclic. Circle is called P-Hagge circle of triangle .
We have a plenty of results about Hagge circle:
(1): Let be the reflections of across the midpoints of , respectively. Prove that are concylic. Note that is the Q-Hagge circle where is the isogonal conjugate point of .
(2): Let again at . Prove that are concurrent.
(3): again at . Prove that are concurrent.
(4): Denote the reflection of across the center of intersect again at . Prove that are concurrent
(5): Given triangle with circumcircle , orthocenter .Denote points on such that the reflections of across the lines , respectively. Prove that are concylic.
(6):Given triangle with circumcircle , orthocenter .Denote points on such that the reflections of across the midpoints of , respectively. Prove that are collinear.
(7): Given triangle and an arbitrary point on circumcircle . Denote be the reflections of across the lines . Prove that are collinear. (Steiner line)
(8): Given triangle and an arbitrary point on circumcircle . Denote be the reflections of across the midpoints of . Prove that are concyclic.
(9): Given triangle and are isogonal conjugate wrt . Let be the circumcenters of P-Hagge’s circle and Q-Hagge’s circle. Prove that
(10): Given triangle , the Nine-point circle and an arbitrary point on . Let be the isogonal conjugate point of wrt . Prove that the center of Q-Hagge circle lies on too. Moreover, is the reflection of across point .
(11): Given triangle and the circumcenter . and are isogonal conjugate wrt triangle such that are collinear. Prove that P-Hagge circle and Q-Hagge circle are tangent.
(12): Given triangle and circumcircle . A circle which has center and radius cuts at , respectively. Let be the Miquel points of triangle wrt and respectively. Prove that E-Hagge circle and F-Hagge circle are congruent.
(13): Denote the incenter, circumcenter, orthocenter and Nagel point of triangle . Prove that
(14): Let be the reflection of wrt the center of P-Hagge, be the centroid of triangle . Prove that are collinear and
(15): When the centroid of triangle . Prove that lies on Q-Hagge.
+ Prove that the center of O-Hagge ( is the circumcenter of triangle )
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