CIRCLE INVESTIGATION:  FINDING THE RELATIONSHIP OF SECANTS, ANGLES AND ARCS – One Case

By

TERESA L. COX

This write-up investigates the relationship of the secants, angles, and arcs of a circle when two secants intersect OUTSIDE the circle.  Other cases will be explored in a later write-up.

First, draw a circle with two secants that intersect outside the circle.  Here is a GSP picture.

We are curious about the relationship of ÐEFD and the arcs that are created by the secants FE and FD.  Is there a relationship between them? Is there a mathematical relationship between ÐEFD, arc GIH, and arc EJD?

At this stage, we know about some relationships about a circle.  Basic facts, such as the diameter is twice the radius, won’t help us here.  The secants do not go through the center A of the circle.  We do know about inscribed angles (see note below), but these secants go outside the circle and do not lie on the circle.

NOTE:  The measurement of an inscribed angle of a circle is 1/2 the measurement of the arc opposite the angle.

As a genus once told me, “Act like a freshman in geometry and connect any points not already connected.”  I want to connect segment HE and create at least one triangle. Here’s the updated picture, with a segment HE and new angles ÐFEH and ÐEHD named Ð1 andf Ð2 respectively.  I will also refer to arc GIH and arc EJD as arc #1 and arc #2 respectively.

We know that inscribed angle Ð1 is 1/2 the measurement of arc #1.  We also know that incscribed angle Ð2 is 1/2 the measurement of arc #2.  Ð2 is an exterior angle to the triangle EFH, so we know that   Ð2 = Ð1 + ÐF.  Let’s restate all these relationships in algebraic form.

Ð2 = Ð1 + ÐF

Ð1 = 1/2 arc #1

Ð2  = 1/2 arc #2

Let’s restate the first equation to solve for ÐF.

ÐF = Ð2 - Ð1

Now substitute what we know about the inscribed angles.

ÐF = 1/2 arc #2 – 1/2 arc #1

Factoring out the 1/;2 to simplify,

ÐF = 1/2 (arc #2 – arc #1)

This is the relationship that exists between the angle and arcs created by the secants of a circle that intersect outside the circle.

Now,  what do you think happens when the secants intersect INSIDE the circle?  That’s another writeup!!!!!!