Đến nội dung

Hình ảnh

toán hình học

- - - - - olympid geometry test

  • Please log in to reply
Chưa có bài trả lời

#1
quannguyenminh103

quannguyenminh103

    Lính mới

  • Thành viên
  • 2 Bài viết

Các anh chị, ai giỏi toán xin vào giúp em với ạ...

Problem  55) Let ABC be a triangle. Incircle (I) touches BC, CA, AB at D, E, F. M is a point on circle center A which passes though E, F.

a) Prove that pedal triangle XYZ of M wwith respect to triangle DEF is right triangle.

b) DM cuts IA at K. MI cuts EF at T. Prove that K lies on circumcircle (DEF) if only if T lies on circumcircle (XYZ).

c) M' is isogonal conjugate of M with respect to triangle DEF. Prove that M' always lies on fixed circle.

 

 

Problem  56) Let ABC be a triangle and point P. A'B'C' is pedal triangle of P with respect to triangle ABC. O is circumcircle of triangles ABC, (O') is circumcircle of triangle A'B'C'. PA', PB', PC' intersects (O') again  at A1, B1, C1, repectively. Assume that P, O O' are collinear. Prove that circumcircles (PAA1), (PBB1), (PCC1) have a common point other than P.

 

 

Problem  57) Let ABC be a triangle with circumcircle (O). A circle (K) pass though B, C intersects AB, AC at F, E, respectively. O1, O2 are circumcenter of triangle ABE, ACF, respectively. (L) is circumcircle of triangle KO1O2. P is point on (L). The line passes though P and perpendiculer to OP intersects (O) at B', C'. Prove that nine-point center of triangle AB'C' always lies on a fixed circle (J) and LJ perpendiculer to EF.







Được gắn nhãn với một hoặc nhiều trong số những từ khóa sau: olympid, geometry, test

2 người đang xem chủ đề

0 thành viên, 2 khách, 0 thành viên ẩn danh