MathDB

3

Part of 2011 Dutch IMO TST

Problems(2)

angle chasing candidate, intersecting circles given, equal angles wanted

Source: Dutch IMO TST1 2011 p3

8/6/2019
The circles Γ1\Gamma_1 and Γ2\Gamma_2 intersect at DD and PP. The common tangent line of the two circles closest to point DD touches Γ1\Gamma_1 in A and Γ2\Gamma_2 in BB. The line ADAD intersects Γ2\Gamma_2 for the second time in CC. Let MM be the midpoint of line segment BCBC. Prove that ∠DPM=∠BDC\angle DPM = \angle BDC.
geometryequal anglesAngle Chasingcircles
three circles given, tangent wanted

Source: Dutch IMO TST2 2011 p3

8/6/2019
Let Γ1\Gamma_1 and Γ2\Gamma_2 be two intersecting circles with midpoints respectively O1O_1 and O2O_2, such that Γ2\Gamma_2 intersects the line segment O1O2O_1O_2 in a point AA. The intersection points of Γ1\Gamma_1 and Γ2\Gamma_2 are CC and DD. The line ADAD intersects Γ1\Gamma_1 a second time in SS. The line CSCS intersects O1O2O_1O_2 in FF. Let Γ3\Gamma_3 be the circumcircle of triangle ADAD. Let EE be the second intersection point of Γ1\Gamma_1 and Γ3\Gamma_3. Prove that O1EO_1E is tangent to Γ3\Gamma_3.
circlestangentgeometry