USING GSP-4 TO EXPLORE RELATIONSHIPS IN CIRCLES
- Open a new sketch. Go to Edit >> Preferences >> Text >> check both For All New Points and As Objects Are Measured; set the Distance units to inches with Precision to hundredths; set Angle Measure to degrees with precision to units.
- Construct a circle with center as point A and the control point as point B. Construct radius AB; use Display >> Line Width >> Thick.
- We want to construct a diameter that is independent of the control point. To do this, use the ray tool to construct a ray that goes from a different point on the circle, C, and passes through the center point A. Place the select tool near the intersection of the circle and the ray to identify and construct the point of intersection, point D. De-select all, then hide the ray; construct the diameter segment CD.
- Select both the radius and diameter; Measure >> Length. Also measure the circumference of the circle.
- Use the calculator to calculate (Circumference ÷ Radius). Did you get the same value as others around you? What happens as you grab-&-drag the center or control point to change the size of the circle?
How do you explain what's going on here? How does this relate to the formula for the circumference (perimeter) of a circle?
- Select the circle; Measure >> Area. What is the formula for the area of a circle? Use the calculator to find the area. To keep the calculation dynamic, instead of using the numerical values, you click on, for example, the measure of length of AB "m segment-AB" will appear in the calculator window.
Better yet, if you select all measures before opening the calculator, then all the measued quantities will appear in Calculate >> Values, along with π.
- Challenge: Can you construct a triangle that has the same perimeter as the circumference of the circle? Which of the figures has the greater area?
Can you construct a quadrilateral that has the same perimeter as the circumference of the circle? Which of the figures has the greater area?
What if you tried this with a pentagon, octagon, 100-gon, etc.? What conjecture might this lead to? How does this relate to those problems that ask to find the greatest area you can enclose with a given length of fencing?
- Construct segment BD. Measure ∠CAB and ∠CDB. How do their measures compare? Does their relationship remain constant even when you dynamically change the size of the circle or the position of the points? How would you summarize this?
- We'd like to construct the minor arc BC. GSP constructs arcs counter-clockwise along a circle. Before constructing an arc, you might want to change the color so you can see the arc on the circle; use Display >> Color. To construct an arc you must select the two endpoints and the circle, then use Construct >> Arc On Circle. Measure >> Arc Angle and also Arc Length.
How does the measure of the arc angle relate to the measures of ∠CAB and ∠CDB?
How does the measure of the arc length relate to the circumference?
- Hide the arc BC and its measures, and hide segment BD. Construct segment BC; change the appearance of the diameter using Display >> Line Width >> Dashed.
Construct the semi-circle arc CD that does not contain point B; Measure both the length and angle of that arc, as well as ∠CBD.
What happens to these measures as you move points B, C and D around? What relationships does this suggest?
Circles are also related to the points of concurrency in triangles. You probably know that the point of concurrency of the angle bisectors is the center of the in-circle. To construct the in-circle you need the center and the end point of a radius. We can use the fact that the radius is perpendicular to a tangent line at the point of tangency to simplify the construction.
Most people do not know about excircles. An excircle is tangent to the extensions of two sides of a triangle and also tangent to the third side. As with the in-circle, we can locate the center and the endpoint of the radius as a point of tangency.
- Construct a generic triangle ABC. Construct the interior angle bisectors from A and B. Label the point of intersection of the bisectors IC for in-center, the center of the in-circle.
- To begin to locate the point of tangency that will determine the radius of the in-circle, construct the line perpendicular to segment AB and passing through point IC. Identify the point of intersection of this perpendicular with segment AB, and label it T1 for first point of tangency. Hide the perpendicular. Select IC and T1, and Construct >> Circle By Center + Point. This is the in-circle.
- Use the ray tool to extend sides AB and AC. Place point P on the extension of ray AB, and point Q on the extension of ray AC. Construct the angle bisectors of ∠PBC and ∠QCB. Their intersection should fall on the bisector of ∠BAC. Label this point of concurrency as XC for ex-center, the center of the excircle.
- To begin to locate the point of tangency that will determine the radius of the excircle, construct the line perpendicular to segment BC and passing through point XC. Identify the point of intersection of this perpendicular with segment BC, and label it T2 for second point of tangency. Hide the perpendicular. Select XC and T2, and Construct >> Circle By Center + Point. This is the excircle.
- Challenge: Locate the other excircle centers. Under what circumstances is the triangle connecting excenters similar to the original triangle?
Other material and challenges concening excircles were found by searching for "excircles" on the Math Forum. For example, from Problems About Excircles:
In fact, you may find a wealth of material on the site Extended Concurrencies of a Triangle.
- Prove that AT1 is the semiperimeter of triangle ABC.
- Prove that triangles like QT2P, determined by the points of tangency of an excircle, are always obtuse.