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 5 of 5: Apply Euler's formula
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OI2=R2−2Rr⟹MNBC=1−2rROI^2=R^2-2Rr\quad\Longrightarrow\quad\dfrac{MN}{BC}=\sqrt{1-\dfrac{2r}{R}}
Detailed analysis

Euler's formula for a triangle is OI squared=R squared minus 2Rr. Since lengths are positive, taking the positive square root in the previous ratio yields MN/BC=square root of (1−2r/R). Equality is an identity for every triangle satisfying the construction.