G.C.A.3 STUDENT NOTES WS #1/#2 – geometrycommoncore.com 3

Construct the circumscribed circle of a triangle (The Circumcircle)

The construction theory of the circumscribed circle of the triangle lies in the relationships found in the perpendicular bisector. So before we get to the specific steps of the construction we need to look at the characteristics found in perpendicular bisectors.

The perpendicular bisector of a segment is the perpendicular line through the midpoint. All points on the perpendicular bisector are equidistant to the two endpoints of the segment. This can easily be proved using congruent triangles and the SAS congruency theorem (common side, right angle, bisected segment).

Recently perpendicular bisectors resurfaced when we looked at the chord properties of a circle. We looked at how the perpendicular bisector of a chord goes through the center of the circle. This is true because the center of the circle must be equidistant to all points on the circle and the two endpoints of a chord are on the circle. We extended this logic as a way to determine the center of a circle. The circle center can be found by constructing the perpendicular bisectors of two chords within a circle. The first perpendicular bisector creates all points equidistant from those two points on the circle and then the second perpendicular bisector intersects the first creating a single location that is equidistant to all four points on the circle – the center.

1. Given a DABC / 2. Create the Perpendicular Bisector of . / 3. Create the Perpendicular Bisector of .
This perpendicular bisector represents all points equidistant
to points A and B. / This perpendicular bisector represents all points equidistant
to points B and C.
4. and because of the perpendicular bisectors and using the transitive property. / 5. Construct the Circumcircle with center D, the circumcenter.
/ The perpendicular bisector of can be created but isn’t necessary because it would go through point D because we have already established that A and C are equidistant to point D.
D is the center of a circle that goes through points A, B and C because they are equidistant to point D.

The location of the circumcenter is sometimes investigated. In a dynamic program like Geometer’s Sketchpad you are able to see that the circumcenter has great mobility such that it can be on, in or out of the triangle.

OUTSIDE – The circumcenter is outside the circle when the triangle is obtuse.
Why would that be?
The obtuse angle is an inscribed angle and so the arc that it subtends
would be greater 180°, a major arc.
This would place the circle center outside the triangle. /
ON – The circumcenter is on the circle when the triangle is a right triangle.
Why would this be?
The right angle is an inscribed angle and so the arc that it subtends would be exactly 180°. An inscribed angle of 90° lies on the diameter of a circle, thus the center would be on the side of the triangle. /
IN – The circumcenter is inside the circle when the triangle is an acute triangle.
Why would this be?
The acute angle is an inscribed angle and so the arc that it subtends
would be less than 180°, a minor arc.
This would place the circle center outside the triangle. /

Construct the inscribed circle of a triangle (The Incircle)

The central construction theory to the inscribed circle is the angle bisector. Its characteristics make this construction possible. Often the characteristic of the angle bisector that we focus on is that it bisects the angles into two congruent angles but in this construction that is secondary to the fact that the angle bisector represents all points that are equidistant to the two sides of the angle. This can be proven using congruent triangles and the AAS congruency theorem (bisected angle, right angle, and common side). The congruent triangles tell us that all points are equidistant to the sides of the angle.

1. Given DABC / 2. Create the Angle Bisector
of ÐCAB. / 3. Create the Angle Bisector
of ÐABC.
The angle bisector represents all points equidistant to
side and side . / This angle bisector represents all points equidistant to
side and side .
4. Point D is equidistant to sides , and . / 5. Determine the distance to a side. Create a perpendicular line to get the perpendicular distance. / 6. Construct the Incircle using the incenter and radius DE.
D is the center of a circle that is equidistant to all three sides of the triangle, the incenter. / The distance DE is same distance from D to and . / , and are
tangent to the incircle.

The incenter never leaves the interior of the triangle.

Prove properties of angles for a quadrilateral inscribed in a circle.

A quadrilateral inscribed in a circle is sometimes called a cyclic quadrilateral. The first thing to notice about cyclic quadrilaterals is that all of their angles are inscribed angles. This is a very important fact in understanding their angle relationships. To find the relationships we need to look at the arcs that are subtended by these inscribed angles.

ÐADC is an inscribed angle on / ÐABC is an inscribed angle on / and form the entire circle, 360°
/ / /
2(mÐADC) + 2(mÐABC) = 360°
2(mÐADC + mÐABC) = 360°
mÐADC + mÐABC = 180°
mÐADC = / mÐABC = / / We learn that the opposite angles of the cyclic quadrilateral are supplementary.
2(mÐADC) = / 2(mÐABC) =
Inscribed angles are half of the arc that they subtend. The opposite inscribed angles of a cyclic quadrilateral subtend together the entire circle, thus together have a value of ½ (360°) = 180°.
They will always be supplementary. / /
mÐADC + mÐABC = 180° / mÐDAB + mÐBCD = 180°