Download problem set #10

Problem Set #10
MATH 387 : 2015
Due: Thursday, 19 March 2015
1. A Lambert quadrilateral is a quadrilateral ABCD with right angles at ∠DAB, ∠ABC, and
∠BCD. Show that the fourth angle ∠CDA is acute, right, or obtuse if and only if the geometry
is semi-hyperbolic, semi-Euclidean, or semi-elliptic respectively.
D
C
C
C
D
D
B
A
B
A
B
A
2. In a semi-hyperbolic or semi-elliptic plane, prove the Angle–Angle–Angle Congruence
Theorem for triangles:
If two triangles ABC and A0 B0C0 satisfy ∠ABC ∼
= ∠A0 B0C0 , ∠BCA ∼
= ∠B0C0 A0 ,
and ∠CAB ∼
= ∠C0 A0 B0 , then the two triangles are congruent.
3. The field of real rational functions
f (t)
g(t) ,
where f (t) and 0 6= g(t) are univariate polynomials
f (t)
an
g(t) > 0 whenever bm > 0
bmt m + am−1t m−1 + · · · + b0 . Arrange the
with real coefficients, can be made into an ordered field by defining
and f (t) = ant n + an−1t n−1 + · · · + a0 and g(t) =
following elements in increasing order:
0,
1,
5,
t,
1
,
t
t + 1,
1
,
t +1
t − 1,
page 1 of 1
t2
,
2
t 2 − t,
t 2 − 1,
1
t+ ,
t
t −1
.
t +1