Segments of Secants and Tangents Theorem

Consider the auxiliary segments $\overline{K L}$ and $\overline{K M} .$

By the Inscribed Angle Theorem, the measure of

∠ M

is half the measure of its intercepted arc

K L .

m ∠ M = \frac{1}{2} m K L

Note that

\overline{J K}

is a tangent to the circle and

\overline{K L}

a chord. Therefore, by the Tangent and Intersected Chord Theorem, it can be said that the measure of

∠ J K L

is half the measure of

K L .

m ∠ J K L = \frac{1}{2} m K L

By the Transitive Property of Equality, it can be stated that

∠ M

and

∠ J K L

have the same measure. Therefore, they are congruent angles. Additionally, by the Reflexive Property of Congruence,

∠ J

is congruent to itself.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ m ∠ M = \frac{1}{2} m K L m ∠ J K L = \frac{1}{2} m K L ⇓ m ∠ M = m ∠ J K L ⇓ ∠ M ≅ ∠ J K L

∠ J ≅ ∠ J

The above information can be visualized in the diagram.

Notice that two angles of

△ J K L

are congruent to two angles of

△ J M K .

Therefore, by the Angle-Angle Similarity Theorem it can be stated that

△ J K L

and

△ J M K

are similar triangles. Consequently, the following proportion can be set.

\frac{J K}{J M} = \frac{J L}{J K}

Lastly, by cross multiplying, the desired result is obtained.

$J K^{2} = J L \cdot J M$

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