Given a triangle ABC, draw lines from each vertex to the opposite side so that all three lines meet.
Label the points on those sides D, E, F.
Label the meeting point O.
See on your drawing that the order of the labeled points on the perimeter is AFBDCEA.

Show that DO/DA + EO/EB + FO/FC = 1 for any triangle.

Ratio of slant heights, DO/DA

            = ratio of altitudes of triangles OBC and ABC on base BC

Thus:    DO/DA = Area of OBC / Area of ABC

and, using similar results,

DO/DA + EO/EB + FO/FC = (sum of areas OBC, OCA, OAB) / area ABC

                                    =  1

Posted by Harry on 2017-07-01 15:34:00