Question

prove that the quadrilateral formed (if possible ) by the internal angle inspectors of any quadrilateral is cyclic ​

Answer 1

Step-by-step explanation:

The average of four numbers is 80.If first three numbers are 80, 90 and 100, find the fourth number.

Answer 2

Answer:

Step-by-step explanation:

Given: ABCD is a quadrilateral AH, BF, CF, DH are bisectors of ∠A, ∠B ∠C ∠D respectively

To prove: EFGH is cyclic quadrilateral

Proof: To prove EFGH is a cyclic quadrilateral, we prove that sum of one pair of opposite angles is 180 degrees.

in triangle AEB

∠ABE + ∠ BAE + ∠AEB = 180 degrees  

∠AEB = 180 degress - ∠ABE - ∠BAE

∠AEB = 180 degress (1/2 ∠B +1/2 ∠A)

∠AEB = 180 degrees -1/2 (∠B + ∠A) ....(1)

Now Lines AH and BF intersect

So ∠ FEH = ∠AEB

∠ FEH = 180 degrees - 1/2 ( ∠B + ∠ A ) ....(2)

Similarly we can prove that  

∠FGH = 180 degrees = 1/2 (∠C + ∠ D).... (3)

addding (2) and (3)

∠FEH + ∠ FGH = 180 degrees -1/2 (∠A + ∠D) +180 degress - 1/2 (∠C +∠ B)

∠FEH + ∠FGH = 180 degrees + 180 Degrees - 1/2 (∠A + ∠D + ∠C + ∠B)

∠ FEH + ∠FGH = 360 degrees - 1/2 (∠A + ∠B + ∠C + ∠D)

∠ FEH + ∠ FGH = 360 degrees -1/2 (∠A + ∠B +∠C + ∠D)

Since ABCD is a quadrilateral

Sum of angles of Quadrilateral = 360 degress

∠A + ∠ B + ∠c + ∠ D = 360 Degrees

∠FEH + ∠FGH = 360 degrees -1/2 x 360 degrees

∠ FEH + FGH = 360 degrees - 180 degrees

∠ FEH + ∠ FGH = 180 degrees

Thus in EFGH

Since sum of one pair of opposite angles is 180 degrees

EFGH is a cyclic quadrilateral