Question

1. Prove Thales’ Theorem: In a circle, if *∠ABC* is an
inscribed angle and AC is a diameter, then
*∠ABC *is a right angle.

Answer #1

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

Given △ABC, extend sides AB and AC to rays AB and AC forming
exterior angles. Let the line rA be the angle bisector ∠BAC, let
line rB be the angle bisector of the exterior angle at B, and let
line rC be the angle bisector of the exterior angle at C.
• Prove that these three rays are concurrent; that is, that they
intersect at a single point. Call this point EA
• Prove that EA is the center of...

Using the hyperbolic angle sum theorem, prove theorem 1. theorem
1 (the exterior angle inequality theorem): An exterior angle of a
triangle is greater than each of the remote/nonadjacent interior
angles of the triangle. hyperbolic angle sum theorem: The sum of
the measures of the angles of a hyperbolic triangle is less than
180°. Please give an original answer, the explanations already
posted on Chegg do not answer this question or do not answer in a
language that I understand...

Find the area of the largest trapezoid that can be inscribed in
a circle of radius 2 and whose base in as diameter of the
circle.

Prove the angle subtraction theorem

If a side of a regular n-gon inscribed in a circle of radius 1
has length x, determine the length of a side of a regular 2n-gon
inscribed in the same circle in terms of x.

prove HL Theorem without angle measure.

1. A square is inscribed in a circle.
a) What is the probability that a point located at random in the
interior of the circle turns out to be also interior to the
square?
b) What is the probability that of 10 points located at random
independently of each other in the interior of the circle, four
fall into the square, three on one segment and one each on the
remaining three segments?

In a circle with center O, consider two distinct diameters: AC
and BD. Prove that the quadrilateral ABCD is a rectangle. Include a
labeled diagram with your proof.

In triangle ABC , let the bisectors of angle b meet AC at D and
let the bisect of angle C meet at AB at E. Show that if BD is
congruent to CE then angle B is congruent to angle C.

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