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 3 of 5: Establish the similarity

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\triangle MNC\sim\triangle IOD

Detailed analysis

The perpendicularities give $\angle ODI=\angle NCM$ . If $\angle ABC=2\alpha$ , the isosceles triangle MBC gives $CM/BC=2\sin\alpha$ . The arc-midpoint construction of D gives $ID/OD=2\sin\alpha$ . Hence $CM/CN=ID/OD$ because $CN=BC$ , and SAS proves that MNC and IOD are similar, with NC corresponding to OD and MN corresponding to IO.