# Equilateral Triangle Inscribed

We will prove that, PQR is an equilateral triangle inscribed in a circle. The tangents at P, Q and R form the triangle P’Q’R’. Prove that P’Q’R’ is also an equilateral triangle.

Solution:

Given:

PQR is an equilateral triangle inscribed in a circle whose centre is O. Three tangents are drawn to the circle at the points P, Q and R. They intersect pair wise at P’, Q’ and R’.

To prove: P’Q’R’ is an equilateral triangle.

Proof:

 Statement Reason 1. ∠Q’PR = ∠PQR.∠Q’RP = ∠PQR. 1. Angle between tangent and chord is equal to the angle in the alternate segment. 2. ∠PQR = 60° 2. Each angle of the equilateral triangle PQR has the measure 60°. 3. ∠Q’PR = ∠Q’RP = 60°. 3. By statements 1 and 2. 4. ∠PQ’R = 180° – (∠Q’PR + Q’RP)             = 180° – (60° + 60°) = 60°. 4. Sum of three angles of a triangle is 180. 5. Similarly, ∠PR’Q = 60°                and ∠QP’R = 60°. 5. Similar reason. 6. ∆P’Q’R’ is equilateral. 6. By statements 4 and 5.

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