Question

Prove that if A*B*C, then ray AB = ray AC and ray BC is a subset of ray AC

Answer #1

Prove that given △ABC and △A′B′C′, if we have AB ≡ A′B′ and
BC≡B′C′,then B<B′ if and only if AC<A′C′. You cannot use
measures.

prove that if C is an element of ray AB and C is not equal to A,
then ray AB = ray AC using any of the following corollarys
3.2.18.) Let, A, B, and C be three points such that B lies on
ray AC. Then A * B * C if and only if AB < AC.
3.2.19.) If A, B, and C are three distinct collinear points,
then exactly one of them lies between the other two.
3.2.20.)...

given a line AB and a point C not on AB, prove that there is a
Ray AD such that AC is between AB and AD

Prove the following identity on languages A, B, C: A(B ∪
C) = AB ∪ AC
Find a counterexample to the following identity on
languages A, B: A* ∩ B* = (A∩B)*

Prove
(a). Events A and B are independent if and only if Ac
and Bc are independent.
(b). If events A and B both have a positive probability and are
disjoint, then they cannot be independent.

1. (a) Let a, b and c be positive integers. Prove that gcd(ac,
bc) = c x gcd(a, b). (Note that c gcd(a, b) means c times the
greatest common division of a and b)
(b) What is the greatest common divisor of a − 1 and a + 1?
(There are two different cases you need to consider.)

A parallelogram ABCD is called a rhombus if AB = BC = CD = DA.
Suppose ABCD is a rhombus. Prove that AC is perpendicular to
BD.

Suppose that the incircle of triangle ABC touches AB at Z, BC at
X, and AC at Y . Show that AX, BY , and CZ are concurrent.

ABC is a right-angled triangle with right angle at A, and AB
> AC. Let D be the midpoint of the side BC, and let L be the
bisector of the right angle at A. Draw a perpendicular line to BC
at D, which meets the line L at point E. Prove that
(a) AD=DE; and
(b) ∠DAE=1/2(∠C−∠B)
Hint: Draw a line from A perpendicular to BC, which meets BC in
the point F

Consider the triangle ABC. Suppose that the perpendicular
bisectors of line segments AB and BC intersect at point X. Prove
that X is on the perpendicular bisector of line segment AC.

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