MathDB

2

Part of 1991 Greece National Olympiad

Problems(3)

2 equidistant points wanted - 1991 Greece MO Grade X p2

Source:

9/6/2024
Let xOy^\widehat{xOy} be an acute angle , AA a point on ray OyOy and BB a point on ray OxOx such that AB⊥OXAB \perp OX .Prove that there are two points on OxOx, each of the equidistant from AA and OxOx.
geometryangle
collinearity and concyclic wanted, 2 intersecting circles

Source: 1991 Greece MO Grade XI p2

9/6/2024
Given two circles (C1)(C_1) and (C2)(C_2) with centers O1\displaystyle{O_1} and O2O_2 respectively, intersecting at points AA and BB. Let ACAC και ADAD be the diameters of (C1)(C_1) and (C2)(C_2) respectively . Tangent line of circle (C1)(C_1) at point AA intersects (C2)(C_2) at point MM and tangent line of circle (C2)(C_2) at point A intersects (C1)(C_1) at point NN. Let PP be a point on line ABAB such that AB=BPAB=BP. Prove that: a) Points B,C,DB,C,D are collinear. b) Quadrilateral AMPNAMPN is cyclic.
geometrycollinearConcyclic
vertors OI = OA_1+OB_1 + OC_1, arc midpoints

Source: 1991 Greece MO Grade XII p2

9/6/2024
Let OO be the circumcenter of triangle ABCABC and let A1,B1,C1A_1,B_1,C_1 be the midpoints of arcs BC,CA,ABBC, CA,AB respectively. If II is the incenter of triangle ABCABC, prove that OI→=OA1→+OB1→+OC1→.\overrightarrow{OI}= \overrightarrow{OA_1}+ \overrightarrow{OB_1}+ \overrightarrow{OC_1}.
geometrycircumcirclevectorarc midpoint