Problem 5

ABC

is a right-angled triangle with right angle at

A

, and

AD

is the altitude to the hypotenuse. The line joining the incenters of

ABD

and

ACD

meets

AB,AC

K,L

. If

S

and

T

are the areas of

ABC

and

AKL

, prove that

S\ge2T

Step 1 of 4: Choose comparison points on the legs

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AK'=AL'=AD

Detailed analysis

Choose K' on AB and L' on AC so that $AK'=AL'=AD$ . Let the perpendicular to AB through K' meet AD at X. Since triangle AK'X and triangle ADB are right triangles with a common angle at A and $AK'=AD$ , they are congruent. Thus the incenter of ADB is also the incenter of AK'X.