Question

In an equilateral triangle ABC; points P, Q and R are taken on the sides AB, BC and CA respectively such that AP= BQ = CR. Prove that triangle PQR is equilateral.

Answer 1

AB = BC = CA…….(i) [Given]

AP = BQ = CR…….(ii) [Given]

Subtracting (ii) from (i)

AB - AP = BC - BQ = CA - CR

BP = CQ = AR …………(ii)

$\angle$A = $\angle$B = $\angle$C……..(iv) [angles opp. to equal sides are equal]
In $\Delta$ BPQ and $\Delta$ CQR
BP=CQ
$\angle$B = $\angle$C
BQ=CR
Frrom SAS $\Delta$BPQ and $\Delta$ CQR are congruent.

_{in $\Delta$ CQR and $\Delta$APR
CQ=AR}
$\angle$_C=$\angle$_{A
CR=AP
From SAS $\Delta$ CQR and $\Delta$ APR are congruent.
so
QR=PR}

From (v) and (vi)

PQ = QR = PR

Therefore, PQR is an equilateral triangle.