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 4 of 4: Use the hypotenuse midpoint

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AM=\frac{BC}{2}\ge AD\quad\Longrightarrow\quad BC\ge2AD\quad\Longrightarrow\quad S\ge2T

Detailed analysis

Let M be the midpoint of BC. In the right triangle ADM, AM is the hypotenuse, so $AM\ge AD$ . Since $BC=2AM$ , this gives $BC\ge2AD$ , and hence $S=[ABC]\ge2[AKL]=2T$ . Equality occurs when D=M, equivalently AB=AC.