MathDB

2

Part of 2016 Iranian Geometry Olympiad

Problems(3)

equal segments concerning circumcircle

Source: IGO Elementary 2016 2

7/22/2018
Let ω\omega be the circumcircle of triangle ABCABC with AC>ABAC > AB. Let XX be a point on ACAC and YY be a point on the circle ω\omega, such that CX=CY=ABCX = CY = AB. (The points AA and YY lie on different sides of the line BCBC). The line XYXY intersects ω\omega for the second time in point PP. Show that PB=PCPB = PC.
by Iman Maghsoudi
geometryequal segmentscircumcircleAngle Chasing
Tangent intersect intersect tangent intersect

Source: Iranian Geometry Olympiad 2016 Medium 2

5/26/2017
Let two circles C1C_1 and C2C_2 intersect in points AA and BB. The tangent to C1C_1 at AA intersects C2C_2 in PP and the line PBPB intersects C1C_1 for the second time in QQ (suppose that QQ is outside C2C_2). The tangent to C2C_2 from QQ intersects C1C_1 and C2C_2 in CC and DD, respectively. (The points AA and DD lie on different sides of the line PQPQ.) Show that ADAD is the bisector of ∠CAP\angle CAP.
Proposed by Iman Maghsoudi
geometry
Iranian Geometry Olympiad (2)

Source: Advanced level,P2

9/13/2016
In acute-angled triangle ABCABC, altitude of AA meets BCBC at DD, and MM is midpoint of ACAC. Suppose that XX is a point such that ∡AXB=∡DXM=90∘\measuredangle AXB = \measuredangle DXM =90^\circ (assume that XX and CC lie on opposite sides of the line BMBM). Show that ∡XMB=2∡MBC\measuredangle XMB = 2\measuredangle MBC.Proposed by Davood Vakili
geometrygeometry proposed