Krishna
0

Step 1: Understand the given question

              GIVEN: A triangle ABC, interior point O.  

                            OD⊥BC , OE⊥AC , and OF⊥AB


Step 2: Join the interior point(O) with the vertices of the triangle (A, B and C)

            CONSTRUCTION:

                

Step 3: According to our requirement we have to select the triangles.

              EXPLANATION: Select the triangles of sides AF, BD and CE. Because,

                                         we have prove the sum of the squares of that sides.

                TRIANGLES:  OFA, OBD and OCE


Step 4: Apply the Pythagoras theorem to the selected right angle triangles.

             EXAMPLE: \triangle OFA

                                 OA^2 = OF^2 + AF^2 ........................(1)

                    Similaly, OB^2 = OD^2 + BD^2 from   \triangle OBD ......................(2)

                   OC^2 = OE^2 + EC^2 from   \triangle OCE .......................(3)


Step 5: Add all the equation to prove the required equaion.

           OA^2 + OB^2 + OC^2 = OF^2 + AF^2 + OD^2 + BD^2 + OE^2 + EC^2

              Arrange this in a particular order  

           OA^2 + OB^2 + OC^2 - OD^2 - OF^2- OE^2 = AF^2 + BD^2 + EC^2

            Hence (i) proved


Step 6: Rewrite the above equation.

            EXAMPLE:   OA^2 + OB^2 + OC^2 - OF^2 - OD^2 - OE^2 = AF^2 + BD^2 + EC^2

             Rewrite the equation in specific manner, which shows the

             Pythagoras formula of the adjacent triangle.

             (OA^2 - OE^2) + (OB^2 - OF^2) + (OC^2 - OD^2) = AF^2 + BD^2 + EC^2

              Replace the equivalent side of the adjacent triangle.

           (AE^2) + (BF^2) + (CD^2) = AF^2 + BD^2 + EC^2

            Hence proved