If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
To prove this theorem we will use the following diagram.
Note that a two-column proof lists each statement on the left and the justification on the right. Each statement must follow logically from the steps before it. In a two-column proof, we always start by stating the given information.
Statement1)& AB and CD are chords
& intersecting in the interior
& of the circle.
Reason1)& Given
To continue our proof, we will write a similarity relation between △ AEC and △ DEB. Therefore, to have △ AEC and △ DEB we need to draw AC and BD.
We can draw these segments by the definition of a segment.
Statement2)& Draw AC and BD.
Reason2)& Definition of a Segment
Now that we have two triangles we can compare them. Notice that ∠ AEC and ∠ DEB are vertical angles.
By the Vertical Angles Congruence Theorem (Theorem 2.6), we can say that these angles are congruent.
Statement3)& ∠ AEC ≅ ∠ DEB.
Reason3)& Vertical Angles
& Congruence Theorem
Now, let's look at ∠ ACD and ∠ ABD. Both of them intercept the same arc, AD.
By the Inscribed Angles of a Circle Theorem (Theorem 10.11) we know that if two inscribed angles of a circle intercept the same arc, then the angles are congruent. Therefore, ∠ ACD and ∠ ABD are congruent.
Statement4)& ∠ ACD ≅ ∠ ABD.
Reason4)& Inscribed Angles of
& a Circle Theorem
With this step, two angles of △ AEC are congruent to the corresponding two angles of △ DEB. From here, by the Angle-Angle (AA) Similarity Theorem (Theorem 8.3) we can conclude that the triangles are similar.
Statement5)& △ AEC ~ △ DEB
Reason5)& AA Similarity Theorem
Since the corresponding side lengths of congruent triangles are proportional, we can write the following proportion.
Statement6)& EA/ED=EC/EB
Reason6)& Corresponding side lengths
& of congruent triangles
& are proportional.
Finally, using the Cross Product Property we will complete the proof.
Statement7)& EB * EA= EC * ED
Reason7)& Cross Product Property
Let's summarize the above process in a two-column table.
Statement
Reason
1.
AB and CD are chords intersecting in the interior of the circle.
1.
Given
2.
Draw AC and BD.
2.
Definition of a Segment
3.
∠ AEC ≅ ∠ DEB
3.
Vertical Angles Congruence Theorem
4.
∠ ACD ≅ ∠ ABD
4.
Inscribed Angles of a Circle Theorem
5.
△ AEC ≅ △ DEB
5.
AA Similarity Theorem
6.
EA/ED=EC/EB
6.
Corresponding side lengths of congruent triangles are proportional.
