# Intersecting Secants Theorem

A theorem is something we can prove in the world of mathematics. One of the most interesting theorems is the intersecting secants theorem. Although this theorem is relatively straightforward, it can be difficult to wrap our minds around at first. Let's take a closer look at this theorem and how it relates to math.

## The intersecting secants theorem defined

Secants are lines that intersect a curve at two points. In the context of circles, if we consider an exterior point outside of a circle, we can draw two secant lines that start from this point and intersect the circle at two points each.

As we can see, the exterior point is M, while the two secant lines are MO and MQ.

So, what exactly is the external secant segments theorem? When we have a situation like this, the theorem states that if we multiply the length of one secant's external segment by the length of the whole secant segment (the sum of the external and internal segments), it will be equal to the product of the lengths of the other secant's external segment and its whole secant segment.

In other words: $\left(\mathrm{MN}\right)*\left(\mathrm{MN}+\mathrm{NO}\right)=\left(\mathrm{MP}\right)*\left(\mathrm{MP}+\mathrm{PQ}\right)$

It is important to note that the external secant segments theorem holds true for any pair of secants, regardless of whether they are symmetrical or not. In fact, we can still use the theorem even if one line merely touches the circle without intersecting it at a second point, forming a tangent.

## Working with the intersecting secants theorem

Let's put our knowledge to good use and practice using the external secant segments theorem:

Take another look at the previous diagram:

Let's assume that we know the following values:

$\mathrm{MN}=10$

$\mathrm{NO}=17$

$\mathrm{MP}=9$

Can we use these values to find the value of PQ?

Let's start by writing out the formula for the intersecting secants theorem:

$\mathrm{MN}*\mathrm{MO}=\mathrm{MP}*\mathrm{MQ}$

Or:

$\left(\mathrm{MN}\right)\left(\mathrm{MN}+\mathrm{NO}\right)=\left(\mathrm{MP}\right)\left(\mathrm{MP}+\mathrm{PQ}\right)$

Now let's substitute in the values that we know:

$\left(10\right)\left(10+17\right)=\left(9\right)\left(9+\mathrm{PQ}\right)$

$\left(10\right)\left(27\right)=\left(9\right)\left(9+\mathrm{PQ}\right)$

$270=81+9\mathrm{PQ}$

At this point, we can subtract 81 from both sides to simplify:

$189=9\mathrm{PQ}$

$21=\mathrm{PQ}$

Congratulations! We have just used the intersecting secants theorem to find the missing value.

## Topics related to the Intersecting Secants Theorem

Intersecting Secant-Tangent Theorem

Angle of Intersecting Secants Theorem

Alternate Exterior Angles