`BC=a;CA=b;AB=c;p=(a+b+c)/2`

Cách `1`

Có `MP////AC=>MP=b/2`

Delta `MXD` cân tại `M`

`=>MX=MD=|CD-CM|=|p-c-a/2|=(|b-c|)/2`

`=>PX=MP+-MX=b/2+-(|b-c|)/2=c/2=PA` hay `DeltaAPX` cân tại `P`

`=>hat(BAX)=hat(BPM)/2=hat(BAC)/2=hat(BAI)=>overline(A,I,X)`

Cách `2`

`AInnBC=G=>(GB)/(AB)=(GC)/(AC)=(BC)/(AB+AC)=a/(b+c)`

`=>GB=(ac)/(b+c);GM=|MB-GB|=|a/2-(ac)/(b+c)|=(a|b-c|)/(2(b+c))`

`=>(GB)/(GM)=((ac)/(b+c))/((a|b-c|)/(2(b+c)))=(2c)/|b-c|

`(XM)/(XP)=((|b-c|)/2)/(c/2)=(|b-c|)/c`

`(AP)/(AB)=1/2`

`=>(AP)/(AB).(XM)/(XP).(GB)/(GM)=1/2.(|b-c|)/c.(2c)/(|b-c|)=1`

`=>overline(A,I,X)` theo Menelaus