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. Hence triangle PQR is equilateral.
State whether the above statement is true or false.

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Solution

Given: ABC is an equilateral triangle. P, Q, R are points on AB, BC, CA respectively, such that
$A P = B Q = C R$
We know, $A B = B C = C A$
$A P = B Q = C R$
$A B - A P = B C - B Q = C A - C R$
$B P = C Q = A R$ (I)
Now, In $△ A P R$ and $△ P B Q$
$∠ A = ∠ B = 60^{\circ}$
$A P = B Q$ (Given)
$B P = A R$ (From I)
Thus, $△ A P R ≅ △ B Q P$ (SAS rule)
Hence, $P Q = P R$ (By cpct)...(II)
Similarly, $△ A P R ≅ △ C R Q$
hence, $P R = Q R$ ..(III)
thus, from II and III
$P Q = Q R = P R$
$∴ △ P Q R$ is an equilateral triangle.

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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. Hence triangle PQR is equilateral.State whether the above statement is true or false.

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. Hence triangle PQR is equilateral.
State whether the above statement is true or false.