Problem 1

Let

ABC

be a triangle with incenter

I

. A point

P

in the interior of the triangle satisfies

\angle PBA+\angle PCA = \angle PBC+\angle PCB

. Show that

AP \geq AI

, and that equality holds if and only if

P=I

Step 4 of 4: Apply the triangle inequality to

A,P,M

Previous step Next step

AP+PM\ge AM=AI+IM\ \Longrightarrow\ AP\ge AI\quad(\text{equality iff }P=I)

Detailed analysis

By the triangle inequality on $A,P,M$ , we have $AP+PM\ge AM=AI+IM$ . Subtracting $PM=IM$ (Step 3) gives $AP\ge AI$ . Equality holds in $AP+PM\ge AM$ if and only if $P$ lies on segment $AM$ ; since segment $AM$ meets the circle centered at $M$ of radius $MI$ only at the single point $I$ , equality holds if and only if $P=I$ .