__Category__: Plane Geometry"Published in Newark, California, USA"

If two circles are tangent externally and a line is drawn through a point of contact and terminated by the circles. Prove that the radii drawn to its extremities are parallel.

Photo by Math Principles in Everyday Life |

__Solution:__

Consider the given figure

Photo by Math Principles in Everyday Life |

Proof:

1. Statement: ∠1

**≅**∠2

Reason: Vertical angles are congruent.

2. Statement: OP

**≅**OA and O'P

**≅**O'B

Reason: All points in a circle are equidistant from its center.

3. Statement: AB is drawn through point P.

Reason: Given item.

4. Statement: ΔOAP and ΔO'PB are isosceles triangles.

Reason: An inscribed triangle in a circle which consist of a center of a circle and the two end points of a chord is always an isosceles triangle.

5. Statement: ∠1

**≅**∠3 and ∠2

**≅**∠4

Reason: The two opposite angles of an isosceles triangle are congruent.

6. Statement: ∠1

**≅**∠2

**≅**∠3

**≅**∠4

Reason: Transitive property of congruence.

7. Statement: ∠AOP = 180º - (∠1 +

**∠3)**

∠PO'B = 180º - (∠2 +

**∠4)**

Reason: The sum of the interior angles of a triangle is 180º.

8. Statement: ∠AOP

**≅**∠PO'B

Reason: By computation at #7, if ∠1

**≅**∠2

**≅**∠3

**≅**∠4, then ∠AOP

**≅**∠PO'B.

9. Statement: OA ║ O'B

Reason: If a transveral line (OO') passed the two alternating interior angles (∠AOP and

**∠PO'B) that are congruent, then it follows that the two lines (OA and O'B) which are adjacent to the alternating interior angles are parallel.**