# 10. Construct a Triangle Given the Length of Its Base, the Difference of the Base Angles and the Slope of the Median to the Base

10. Construct a Triangle Given the Length of Its Base, the Difference of the Base Angles and the Slope of the Median to the Base

This Demonstration shows a construction of a triangle given the length of the base , the difference of the base angles and the slope of the median from to the base.

ABC

c

AB

δ

ϵ

m

C

C

Construction

Draw a segment of length with midpoint .

AB

c

E

Step 1: Draw a line through with slope relative to . Choose any point on . Let be the intersection of the perpendicular bisector to and the ray through that forms an angle with the perpendicular from to .

λ

E

ϵ

AB

C'

λ

S

AB

C'

δ

C'

AB

Step 2: Draw a circle with center and radius . Let and be intersections of and .

σ

S

SC'

A'

B'

σ

AB

Step 3: Let be the intersection of and the line through parallel to .

C

λ

A

A'C'

Verification

Let and .

∠CAB=α

∠ABC=β

In any triangle, the angle between the altitude to the base and the segment , where is the circumcenter, is .

CS

S

α-β

By construction, the triangles and are similar, so . The line segment is the median of with slope since is on and is the midpoint of .

ABC

A'B'C'

δ=α-β

CE

ABC

ϵ

C

λ

E

AB