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In quidrilateral $$ABCD$$, The diagonals $$\overline{AC}$$ and $$\overline{BD}$$ intersect at the point $$O$$, and the following four sets of conditions are given:

①$$\overline{AB}\parallel\overline{CD},\overline{AD}\parallel\overline{BC}$$;

②$$AB = CD,AD = BC$$;

③$$AO = CO,BO = DO$$;

④$$\overline{AB}\parallel\overline{CD},AD = BC$$.

The number of sets which must be able to deduce that this quadrilateral is a parallelogram is ( ).

In quadrilateral $$ABCD$$, diagonals $$\overline{AC}$$ and $$\overline{BD}$$ intersect at point $$O.m\angle ADB = m\angle CBD$$. Which of the following conditions, when added, still cannot deduce that $$ABCD$$ is a parallelogram?

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As shown in the figure, in $$\triangle ABC$$, $$D$$ and $$E$$ are the mid - points of $$\overline{AB}$$ and $$\overline{BC}$$, respectively. $$\overline{DE}\parallel\overline{AC}$$, $$DE = \frac{1}{2}AC$$ and point $$F$$ is on $$\overline{DE}$$. If a condition is added to make the quadrilateral $$ADFC$$ a parallelogram, then this condition is _____.

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As shown in the figure, in parallelogram $$ABCD$$, diagonals $$\overline{AC}$$ and $$\overline{BD}$$ intersect at point $$O$$. $$E$$ and $$F$$ are two points on $$\overline{AC}$$. If $$E$$ and $$F$$ satisfy one of the conditions below, then quadrilateral $$DEBF$$ is not necessarily a parallelogram. This condition is ( ).

（含图）

Rectangle $$ABCD$$ has $$AB = 6$$ and $$BC = 3$$. Point $$M$$ is chosen on side $$AB$$ so that $$m\angle AMD=m\angle CMD$$. What is the degree measure of $$\angle AMD$$? (2011 AMC 10B Problem, Question #18)

Two equilateral triangles are contained in a square whose side length is $$2\sqrt{3}$$. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus? (2012 AMC 10B Problem, Question #10)

As shown in the figure, in rhombus $$ABCD$$, $$m\angle A = 110^{\circ}$$, points $$E$$ and $$F$$ are mid - points of $$\overline{AB}$$ and $$\overline{BC}$$, respectively. $$\overline{EP}\perp\overline{CD}$$ at point $$P$$. Then $$m\angle PEF=$$_____. (缺图)

Trapezoid $$ABCD$$ has $$\overline{AD}\parallel\overline{BC}$$, $$BD = 1$$, $$m\angle DBA = 23^{\circ}$$, and $$m\angle BDC = 46^{\circ}$$. The ratio $$BC:AD$$ is $$9:5$$. What is $$CD$$? (2009 AMC12B Problem, Question #16) (缺图)

Trapezoid $$ABCD$$ has parallel sides $$\overline{AB}$$ of length $$33$$ and $$\overline{CD}$$ of length $$21$$. The other two sides are of lengths $$10$$ and $$14$$. The angles at $$A$$ and $$B$$ are acute. What is the length of the height of $$ABCD$$?(Adapted from 2014 AMC 10B Problem, Question #21)

（含图）

As shown in the figure, which of the following statements cannot be used to deduce that $$\triangle ADB\sim\triangle ABC$$? ( )

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In the figure, $$\angle EAB$$ and $$\angle ABC$$ are right angles. $$AB = 4, BC = 5, AE = 10$$, and $$\overline{AC}$$ and $$\overline{BE}$$ intersect at $$D$$. What is the difference between the areas of $$\triangle ADE$$ and $$\triangle BDC$$? (Adapted from 2004 AMC 10A Problem, Question #9) (缺图)

In rectangle $$ABCD$$, $$AB = 6$$ and $$BC = 8$$. Point $$E$$ between $$B$$ and $$C$$, and point $$F$$ between $$E$$ and $$C$$ are such that $$BE = EF = FC$$. Segments $$\overline{AE}$$ and $$\overline{AF}$$ intersect $$\overline{BD}$$ at $$P$$ and $$Q$$, respectively. The ratio $$BP:PQ:QD$$ can be written as $$r:s:t$$ where the greatest common factor of $$r$$, $$s$$ and $$t$$ is $$1$$. What is $$r + s + t$$? (2016 AMC 10A Problem, Question #19) (缺图)

Rectangle $$ABCD$$ has $$AB = 8$$ and $$BC = 6$$. Point $$M$$ is the mid - point of diagonal $$\overline{AC}$$, and $$E$$ is on $$\overline{AB}$$ with $$ME\perp AC$$. What is the area of $$\triangle AME$$? (2009 AMC 10B Problem, Question #18) (缺图)

Among the following statements: ① Diameters are chords, but chords are not necessarily diameters; ② Semicircles are arcs, but arcs are not necessarily semicircles. ③ Two circles of the same radius are congruent circles. ④ A chord divides the circle into two arcs, at least one of them is a major arc, there are ____ correct statements. (缺图)

The number of correct statements is ____. ① A semicircle is an arc; ② An arc is a semicircle; ③ A diameter is a chord; ④ Arcs with the same length are congruent arcs; ⑤ A diameter divides the circle into two arcs, each of them is a semicircle. (缺图)

As shown in the figure, $$\overrightarrow{AD}$$ is the diameter of $$\odot O$$. $$\overrightarrow{AD}\perp\overrightarrow{BC}$$ at $$E$$. If $$DE = 3$$, and $$BC = 8$$, then the radius of $$\odot O$$ is ____. (缺图)

$$P$$ is a point inside a circle $$O$$ whose radius is 4. If $$OP = 3$$, then what is the number of chords that pass through point $$P$$ and have an integer length? (缺图)

A square of side length 1 and a circle of radius $$\frac{\sqrt{3}}{3}$$ share the same center. What is the area inside the circle, but outside the square? (2010 AMC 10B Problem, Question #16) (缺图)

Let $$\overrightarrow{AB}$$ be a diameter in a circle of radius $$5\sqrt{2}$$. Let $$\overrightarrow{CD}$$ be a chord in the circle that intersects $$\overrightarrow{AB}$$ at a point $$E$$ such that $$BE = 2\sqrt{5}$$ and $$m\angle AEC = 45^{\circ}$$. What is $$CE^{2}+DE^{2}$$? (2020 AMC 12B Problem, Question #12) (缺图)

Line $$l$$ intersects $$\odot O$$, whose radius is $$r$$, and the distance from point $$O$$ to line $$l$$ is 6. Then the range of $$r$$ is ____. (缺图)

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