6. Construct a Triangle Given Its Circumradius, Inradius and the Length of Its Base
6. Construct a Triangle Given Its Circumradius, Inradius and the Length of Its Base
This Demonstration constructs a triangle given the circumradius , inradius and the length of the base .
ABC
R
r
AB=c
Construction
Draw a circle of radius and a chord of length .
σ
1
R
AB
c
Step 1: Let be the midpoint of and let the right bisector of meet at the point .
D
AB
AB
σ
1
E
Step 2: Draw a second circle with center through and . Draw a (green) line perpendicular to at so that . Clearly is parallel to .
σ
2
E
A
B
p
ED
F
DF=r
p
AB
Step 3: Let be one of the points of intersection of and . The point is the intersection of and the ray .
G
p
σ
2
C
σ
1
EG
Verification
Let , and .
∠CAB=α
∠ABC=β
∠BCA=γ
Since arc equals arc , bisects and . Also . The triangle is isosceles, so . Then . So is the incenter of with distance from .
AE
EB
EC
∠ACB
∠BAE=γ/2
∠AEG=∠ABC=β
AEG
∠EAG=π/2-β/2
∠GAB=∠GAE-∠BAE=π/2-β/2-γ/2=α/2
G
ABC
r
AB