Download Construct the circumscribed circle of a triangle

G.C.A.3 STUDENT NOTES WS #1/#2 – geometrycommoncore.com
1
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.
B
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).
A
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 ABC
2. Create the Perpendicular
Bisector of AB .
3. Create the Perpendicular
Bisector of BC .
B
B
B
C
C
A
A
A
This perpendicular bisector
represents all points equidistant
to points A and B.
4. AD  BD and BD  CD
because of the perpendicular
bisectors and AD  BD  CD
using the transitive property.
B
C
A
D
D is the center of a circle that goes
through points A, B and C because
they are equidistant to point D.
This perpendicular bisector
represents all points equidistant
to points B and C.
5. Construct the Circumcircle with
center D, the circumcenter.
B
C
C
A
D
The perpendicular bisector of AC
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.
G.C.A.3 STUDENT NOTES WS #1/#2 – geometrycommoncore.com
2
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.
B
C
A
D
B
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.
C
D
A
B
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.
D
C
A
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
B
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 ABC
2. Create the Angle Bisector
of CAB.
B
B
C
A
3. Create the Angle Bisector
of ABC.
B
C
A
The angle bisector represents all
points equidistant to
side AB and side AC .
C
A
This angle bisector represents all
points equidistant to
side AB and side BC .
G.C.A.3 STUDENT NOTES WS #1/#2 – geometrycommoncore.com
4. Point D is equidistant to sides
AB , BC and AC .
3
5. Determine the distance to a
side. Create a perpendicular line
to get the perpendicular distance.
B
6. Construct the Incircle using the
incenter and radius DE.
B
D
B
D
C
A
E
A
D is the center of a circle that is
equidistant to all three sides of the
triangle, the incenter.
D
C
A
The distance DE is same distance
from D to BC and AB .
C
E
AB , BC and AC 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.
B
A
E
C
D
ADC is an inscribed
ABC is an inscribed
angle on ABC
angle on AC
B
ABC and AC form
the entire circle, 360
B
mABC  mAC  360
B
2(mADC) + 2(mABC) = 360
A
A
E
C
D

1
m ABC
2

2(mADC) = mABC
mABC =
(mADC + mABC) = 360
E
C
D
mADC =

A
E
C
mADC + mABC = 180
D

1
m AC
2

We learn that the opposite angles
of the cyclic quadrilateral are
supplementary.
mABC  mAC  360
2(mABC) = mAC
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.
B
B
A
A
E
E
C
C
D
mADC + mABC = 180
D
mDAB + mBCD = 180