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