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.
A proposed highway has two tangents of bearings N
45°54 ,36,, E and N
1°22 ,30,, W. The highway
design engineer, attempting to obtain the best fit for the simple
circular curve to join these tangents, decides that the external
ordinate is to be 43.00 ft. The PI is at station 65+ 43.21
Determine:
(a) The central angle of the curve
(b) The radius of the curve
(c) The length of the tangent of the curve
(d) The station of...