# 1. Construct a Triangle Given the Length of the Base, the Difference of the Base Angles and the Foot of the Altitude to the Base

1. Construct a Triangle Given the Length of the Base, the Difference of the Base Angles and the Foot of the Altitude to the Base

This Demonstration constructs a triangle given the length of its base, the difference of the base angles and the foot of the altitude from to the base.

AB=c

δ

D

C

Construction

Step 1: Draw a horizontal line of length and the foot of the altitude from . Measure out a point so that , so that is the midpoint of .

AB

c

D

C

E

ED=AD

D

AE

Step 2: Draw a vertical line through .

λ

D

Step 3: Draw a circle with center so that the central angle .

σ

S

∠ESB=2δ

Step 4: Let the intersections of and , if they exist, be and . Join , , and .

σ

λ

C

1

C

2

AC

1

AC

2

BC

1

BC

2

Then either of the two triangles or is a solution.

ABC

1

ABC

2

Step 5: Draw the triangle E (dashed line).

C

1

C

2

Verification

Let AB=α and =β.

∠C

1

∠ABC

1

Consider the triangle (the verification for the triangle is similar). Because is the right bisector of , =α, so the angle B=α-β. But this angle subtends the chord and is half of the central angle over , which is . So .

ABC

1

ABC

2

λ

AE

∠DEC

1

∠EC

1

EB

EB

2δ

δ=α-β