MathLabs

Problem 5

In triangle ABC, points M and N lie on AB and AC, respectively, and MB=BC=CN. If R and r are the circumradius and inradius of ABC, express MN/BC in terms of R and r.
Step 1 of 5: Choose the arc midpoint
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D=BI∩Γ,D≠B,OD⊥ACD=BI\cap\Gamma,\quad D\ne B,\quad OD\perp AC
Detailed analysis

Let I and O be the incenter and circumcenter, and let D be the second point where BI meets the circumcircle. D is the midpoint of arc AC, so OD is perpendicular to chord AC and hence to CN.