Problem 5

Let

I

be the incenter of triangle

ABC

. Let the incircle of

ABC

touch the sides

BC

CA

, and

AB

K

L

, and

M

, respectively. The line through

B

parallel to

MK

meets the lines

LM

and

LK

R

and

S

, respectively. Prove that angle

RIS

is acute.

Step 1 of 4: Identify the key perpendicular

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In plain words

The incenter is the center of the incircle.

BI\perp MK\parallel RS

Detailed analysis

Because $M$ and $K$ are the endpoints of a chord of the incircle and $I$ is its center, $BI$ is the perpendicular bisector of $MK$ . Thus $BI\perp MK$ , and the line $RS$ through $B$ is also perpendicular to $BI$ .