MathDB

3

Part of 2018 Saudi Arabia GMO TST

Problems(2)

circumcenter of XYZ = circumcenter of A_1B_1C_1 iff incenter = orthocenter

Source: 2018 Saudi Arabia GMO TST I p3

8/1/2020
Let I,OI, O be the incenter, circumcenter of triangle ABCABC and A1,B1,C1A_1, B_1, C_1 be arbitrary points on the segments AI,BI,CIAI, BI, CI respectively. The perpendicular bisectors of AA1,BB1,CC1AA_1, BB_1, CC_1 intersect each other at X,YX, Y and ZZ. Prove that the circumcenter of triangle XYZXYZ coincides with OO if and only if II is the orthocenter of triangle A1B1C1A_1B_1C_1
geometryincenterCircumcenterorthocenter
perpendicular wanted, arc midpoint, tangent lines to circles

Source: 2018 Saudi Arabia GMO TST II p3

8/1/2020
Let CC be a point lies outside the circle (O)(O) and CS,CTCS, CT are tangent lines of (O)(O). Take two points A,BA, B on (O)(O) with MM is the midpoint of the minor arc ABAB such that A,B,MA, B, M differ from S,TS, T. Suppose that MS,MTMS, MT cut line ABAB at E,FE, F. Take X∈OSX \in OS and Y∈OTY \in OT such that EX,FYEX, FY are perpendicular to ABAB. Prove that XYX Y and CMC M are perpendicular.
geometryperpendicularTangentsarc midpoint