MathLabs

Problem 5

ABCABC is a right-angled triangle with right angle at AA, and ADAD is the altitude to the hypotenuse. The line joining the incenters of ABDABD and ACDACD meets AB,ACAB,AC at K,LK,L. If SS and TT are the areas of ABCABC and AKLAKL, prove that S≥2TS\ge2T.
Step 3 of 4: Compute the two areas
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[ABC]=BC⋅AD2,[AKL]=AD22[ABC]=\frac{BC\cdot AD}{2},\qquad [AKL]=\frac{AD^2}{2}
Detailed analysis

Since K=K' and L=L', we have [AKL]=AD2/2[AKL]=AD^2/2. Also [ABC]=BC⋅AD/2[ABC]=BC\cdot AD/2. Thus [ABC]≥2[AKL][ABC]\ge2[AKL] is equivalent to BC≥2ADBC\ge2AD.