prove that a triangle inscribed in a circle is an equilateral triangle. given arcPQ=arcQR=arcRP. prove that trianglePQR is an equilateral triangle

Question

need help. just some simple proving that's all. tnx

Answer 1

 Let the centre of the circle be O..

Join all the points p, q, r to the center(O) of the circle.

As it is given arcs arcPQ=arcQR=arcRP ,lines OP,OQ,OR divides the circle into three equal sectors,with all equal angles.

Now take triangle POQ and triangle ROQ

            side PO = side OR

            side OQ is common

             <POQ  = <ROQ

  So, triangle POQ and triangle ROQ are congruent 

so by CPCT property , PQ = RQ

 

 

Similarily we can prove that triangles POQ and triangles POR are congruent and so PQ = PR

 

 

therefore PQ=PR=RQ so the triangle PQR will be an equilateral triangle..

Answer 2

arc PQ=arc QR    Given

<R=<P Angle-Arc Theorem  (<R=1/2 measure of arc PQ and <P= 1/2 measure of arc QR)

arc QR=arc RP    Given

<P=<Q Angle-Arc Theorem  (<P= 1/2 measure of arc QR and <Q= 1/2 measure of arc RP)

if <R=<P and <P=<Q, then  <R=<Q     Transitive Property

triangle PQR is an equilateral triangle because all three angles are congruent (<R=<P=<Q)