Question

in the fig D, E, F are midpoints of sides AB, BC and AC respectively. P is the foot of the perpendicular from A to side BC. Show that points D, F, E and P are concyclic​

Answer 1

Answer:

Step-by-step explanation:

Given : In Δ ABC , D,E,F are the mid _ point of the sides BC , CA, AND AB  respectively , AP ⊥ BC

TO prove : D,F,E and P are con cyclic.

Proof : In right angled triangle  APB, D is the mid point of AB

                        DB =DP

                 ∠ 2 = ∠1 ....(1)

( angle opposite to the equal sides are equal)

Since D and F are the mid point of AB and AC , then

                                         DF║ BC

                                        DF ║ BE

Since EF ║DB , then quadrilateral BEFD is a parallelogram

                                                 ∠1 = ∠3....(2)

 From equation 1 & 2

                                          ∠2 =∠3

                But              ∠2 + ∠4 = 180 °         ( linear pair Axiom )

∴                                   ∠3 + ∠4 = 180°              (∵ ∠2 =∠3 )

hence , Quadrilateral EFDP is cyclic quadrilateral

  So, points D,E,F and P are con - cyclic