MathDB

6

Part of Mathley 2014-15

Problems(2)

line through N perpendicular to PC crosses a fixed point as P moves along EF

Source: Mathley 2014.1 p6

8/19/2020
Let the inscribed circle (I)(I) of the triangle ABCABC, touches CA,ABCA, AB at E,FE, F. PP moves along EFEF, PBPB cuts CACA at M,MIM, MI cuts the line, through CC perpendicular to ACAC, at NN. Prove that the line through NN is perpendicular to PCPC crosses a fixed point as PP moves.
Tran Quang Hung, High School of Natural Sciences, Hanoi National University
fixedFixed pointgeometryincircle
there is a circle with center O tangent to all 4 circumcircles , bicentric

Source: Mathley 2014.3 p6

8/19/2020
A quadrilateral is called bicentric if it has both an incircle and a circumcircle. ABCDABCD is a bicentric quadrilateral with (O)(O) being its circumcircle. Let E,FE, F be the intersections of ABAB and CD,ADCD, AD and BCBC respectively. Prove that there is a circle with center OO tangent to all of the circumcircles of the four triangles EAD,EBC,FAB,FCDEAD, EBC, FAB, FCD.
Nguyen Van Linh, a student of the Vietnamese College, Ha Noi
geometrycircumcirclebicentric quadrilateraltangent circles