Question

prove that the quadilateral formed by bisectors of angles of a quadilateral is cyclic

Answer 1

ABCD is a cyclic quadrilateral ∴∠A +∠C = 180∘ and ∠B+ ∠D =  180∘

12∠A+12 ∠C = 90∘ and 12 ∠B+12 ∠D =  90∘

x + z = 90∘ and y + w =  90∘

In △ARB and △CPD, x+y + ∠ARB = 180∘ and z+w+ ∠CPD =  180∘

∠ARB = 180∘ – (x+y) and ∠CPD = 180∘ – (z+w)

∠ARB+∠CPD = 360∘ – (x+y+z+w) = 360∘ – (90+90)

                        = 360∘ – 180∘ ∠ARB+∠CPD =  180∘

∠SRQ+∠QPS =  180∘

The sum of a pair of opposite angles of a quadrilateral PQRS is 180∘.

Hence PQRS is cyclic

 

Answer 2

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