Prove the following theorem:
In a Pasch geometry, a quadrilateral is a convex quadrilateral
if and...
Prove the following theorem:
In a Pasch geometry, a quadrilateral is a convex quadrilateral
if and only if the vertex of each angle is contained in the
interior of the opposite angle.
Prove the following using the triangle inequality:
Given a convex quadrilateral, prove that the point determined by
the intersection of the diagonals is the minimum distance point for
the quadrilateral - that is, the point from which the sum of the
distances of the vertices is minimal.
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.
Explain what it is a neutral theorem
in Euclidean geometry.
State & prove both: the theorem on construction of parallel
lines and its converse. Which one of them is neutral?
Considering the illustrations for Concave and Convex
mirrors.
Prove using geometry that the reflected rays reach the focal
point f=R/2 in the limit as the incoming rays approach the
principal axis.
Hint: Consider the triangle formed by the radius of curvature,
principal axis, and reflected ray, and use the law of sines.
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.
The goal of this exercise is to prove the following theorem in
several steps.
Theorem: Let ? and ? be natural numbers. Then, there exist
unique
integers ? and ? such that ? = ?? + ? and 0 ≤ ? < ?.
Recall: that ? is called the quotient and ? the remainder of the
division
of ? by ?.
(a) Let ?, ? ∈ Z with 0 ≤ ? < ?. Prove that ? divides ? if and...
Prove that if two of the opposite sides of a quadrilateral are
respectively the greatest and the least sides of the quadrilateral,
then the angles adjacent to the least are greater than their
opposite angles.
Prove that if two of the opposite sides of a quadrilateral are
respectively the greatest and the least sides of the quadrilateral,
then the angles adjacent to the least are greater than their
opposite angles
For each of the following sets, prove that thay are convex sets
or not. Also graph the sets.
a) ? 1= {(?1 , ?2 ): ?1 ^2 + ?2^2 ≥ 1}
b)?2 = {(?1 ,?2 ): ?1 ^2 + ?2^ 2 = 1}
c)?3 = {(?1 , ?2 ): ?1 ^2 + ?2 ^2 ≥ 1}