# 2. Construct a Triangle Given the Circumradius, the Difference of the Base Angles, with the Circumcenter on the Incircle

2. Construct a Triangle Given the Circumradius, the Difference of the Base Angles, with the Circumcenter on the Incircle

This Demonstration shows a construction of a triangle given its circumradius , the difference of the base angles and that the circumcenter is on the incircle.

ABC

R

δ

Let be the inradius of . The Euler formula =R(R-2r) gives the distance between the circumcenter and the incenter. Since the circumcenter is on the incircle, =R(R-2r), which has the positive solution .

r

ABC

2

z

z

2

r

r=R(1+

2

)Construction

Draw a circle of radius with center and draw a diameter . Draw a chord at an angle from .

σ

1

R

S

CF

CD

δ/2

CF

Step 1: Draw a circle with center and radius . Of the two points of intersection of and the segment , let the point be the one closest to .

σ

2

S

r

σ

2

CD

E

C

Step 2: Draw a ray from at an angle from . Let be the perpendicular projection of on . Measure out a point on at distance from .

ρ

C

δ

CF

G

E

ρ

H

ρ

r

G

Step 3: The points and are the intersections of and the line through is perpendicular to .

A

B

σ

1

H

ρ

Verification

Let , and .

∠CAB=α

∠ABC=β

∠BCA=γ

Theorem: Let be any triangle. Let be the foot of the altitude from to , and let be the center of the circumscribed circle. Then the angle at between the altitude and equals . The angle between and the angle bisector at is . (See The Plemelj Construction of a Triangle 4.)

ABC

H

C

AB

S

C

CH

CS

α-β

CH

C

(α-β)/2

Proof of the last part: Let be on the angle bisector at , then .

E

C

∠HCE=∠ACE-∠ACH=γ/2-(π/2-α)=π/2-α/2-β/2-π/2+α=(α-β)/2

By construction and the theorem, and is the circumscribed circle of triangle with center and radius .

δ=α-β

σ

1

ABC

S

R

By construction, is on the angle bisector at and the distance of to is . So the circle with center and radius is the incircle of , which by construction contains .

E

C

E

AB

r

E

r

ABC

S