Let E be a projective conic through the quadrilateral ABCD, and
let the tangents to E...
Let E be a projective conic through the quadrilateral ABCD, and
let the tangents to E at A and C meet at P on the Line BD. Show
that the tangents to E at B and D meet at a Point Q on AC.
Theorem (Three Tangents): Let a non-degenerate plane conic touch
the sides BC, CA, and AB of a triangle ABC in R2 at the
points P, Q, and R respectively. Then AP, BQ, and CR are
concurrent.
Please provide a proof of the Three Tangents Theorem without
reference to Ceva's Theorem.
Hint: Consider the Three Point Theorem
Draw a convex quadrilateral ABCD, where the diagonals intersect
at point M. Prove: If ABCD is a parallelogram, then M is the
midpoint of each diagonal.
In hyperbolic geometry, suppose ABCD is a quadrilateral with
right angles at C and D such that AD = BC. Show that AB > CD.
Hint: Use Proposition 24 of Euclid.
State the dual of the Theorem below.
Let a non-degenerate plane conic touch the sides BC, CA, and AB
of a triangle ABC in R2 at the points P, Q, and R
respectively. Then AP, BQ, and CR are concurrent.
Q2. Let (E, d) be a metric space, and let x ∈ E. We say that x
is isolated if the set {x} is open in E.
(a) Suppose that there exists r > 0 such that Br(x) contains
only finitely many points. Prove that x is isolated.
(b) Let E be any set, and define a metric d on E by setting d(x,
y) = 0 if x = y, and d(x, y) = 1 if x and y...
Let E/F be a field extension. Let a,b be elements
elements of E and algebraic over F. Let m=[Q(a):Q] and n=[Q(b):Q].
Assume that gcd(m,n)=1. Determine the basis of Q(a,b) over Q.