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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