seg AD perpendicular side BC,seg BE perpendicular side AC,seg CF perpendicular side AB. point O is the othocentre. prove that, point O is the incentre of tiangle DEF .
Answers
Yes. Draw the figure. Because angles BFC and BEC are right, points B, E, F, C all lie on a circle of diameter BC. Therefore angles FEB and FCB are equal, as they subtend the same arc BF on that circle. Likewise, because angles ADB and AEB are right, points A, E, D, B all lie on a circle of diameter AB. Therefore angles BED and BAD are equal, as they subtend the same arc AB on that circle.
But angles FCB and BAD are equal, being supplementary to angle B. So angles FEB and BED are also equal. But thus means that BE is the interior bisector of angle FED. Similarly, AD and CF are the bisectors of angles EDF and DFE respectively. So their common point O is the incenter of triangle DEF.
You need a different figure if triangle ABC is obtuse-angled, but a similar reasoning can be carried through.
