# Two Tangents are Drawn to a Circle from an External Point

We will prove that the tangents MX and MY are drawn to a circle with centre O from an external point M. Prove that ∠XMY = 2∠OXY.

Solution:

Proof:

 Statement Reason 1. In ∆MXY, MX = MY. 1. Tangents drawn to a circle from an external point are equal. 2. ∠MXY = ∠MYX = x°. 2. From statement 1. 3. ∠XMY = 180° - x°. 3.Sum of three angles of a triangle is 180°. 4. OX ⊥ XM, i.e., ∠OXM = 90°. 4. Radius ⊥ tangent. 5. ∠OXY = 90° - ∠MXY⟹ ∠OXY = 90° - x°⟹ 2∠OXY = 180° - 2x° 5. From the figure. 6. Therefore, ∠XMY = 2∠OXY. (Proved). 6. From statements 3 and 5.

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