Question

Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any quadrilateral is cyclic.

Answer 1

Hint: A quadrilateral is a polygon having four sides and four vertices. A quadrilateral is cyclic when all of its vertices lie on a circle.
A cyclic quadrilateral is given by the following figure:
 

A quadrilateral formed by the internal angle bisectors of a quadrilateral ABCD is PQRS:

We need to prove PQRS is cyclic.
A rule of angle equality is, vertically opposite angles are equal.
Angle sum property of a triangle gives the sum of all angles of a triangle = ${180^0}$
To prove the quadrilateral PQRS is cyclic, it is enough to prove that the sum of opposite angles of PQRS is ${180^0}$ . Implies $\angle SPQ + \angle SRQ = {180^0}$

Complete step-by-step answer:
Step 1: Consider a quadrilateral ABCD with internal bisectors AQ, BS, CS, DQ of angles $\angle A$, $\angle B$ , $\angle C$ and $\angle D$ respectively. We need to prove the quadrilateral PQRS formed by these four internal angle bisectors is cyclic.
As vertically opposite angles are equal, we obtain the following equalities:
$
  \angle SPQ = \angle APB \\
  \angle SRQ = \angle DRC \\
 $
Adding above two formulas we get $\angle SPQ + \angle SRQ = \angle APB + \angle DRC$
As AQ is the angle bisector of $\angle A$ and P is a point on AQ, By angle sum property,
 $\angle APB = 180 - (\dfrac{1}{2}\angle A + \dfrac{1}{2}\angle B)$ and $\angle DRC = 180 - (\dfrac{1}{2}\angle D + \dfrac{1}{2}\angle C)$ . Thus,
 $
  \angle SPQ + \angle SRQ = \angle APB + \angle DRC \\
   = 180 - (\dfrac{1}{2}\angle A + \dfrac{1}{2}\angle B) + 180 - (\dfrac{1}{2}\angle D + \dfrac{1}{2}\angle C) \\
   = 360 - \dfrac{1}{2}(\angle A + \angle B + \angle C + \angle D) \\
   = 360 - \dfrac{1}{2}(360) \\
   = {180^0} \\
 $
The sum of opposite angles of quadrilateral PQRS is ${180^0}$ . Thus PQRS is a cyclic quadrilateral.
Hence proved.
The quadrilateral formed by internal angle bisectors of a quadrilateral is cyclic.

Note: Students should always draw a diagram for better understanding of such questions. Also, they should avoid writing the angles in one letter when taking in consideration a figure where there are several other angles which might seem like the same.
Kite, Trapezoid, Parallelogram, Square, Rhombus, Rectangle comes under Quadrilateral.