英语翻译1． Two planes,and ,intersect along the line .The point is given in the plane ,and the point in the plane ; neither of these points lies on the straight line .Construct an isosceles trapezoid (with ) in which a circle can be inscribed,an

英语翻译1． Two planes,and ,intersect along the line .The point is given in the plane ,and the point in the plane ; neither of these points lies on the straight line .Construct an isosceles trapezoid (with ) in which a circle can be inscribed,an
英语翻译
1． Two planes,and ,intersect along the line .The point is given in the plane ,and the point in the plane ; neither of these points lies on the straight line .Construct an isosceles trapezoid (with ) in which a circle can be inscribed,and with vertices and lying in planes and respectively
2． Construct triangle ,given ,(the altitudes from and ),and ,the median from vertex .
3． Construct a triangle if ,and ,where is the midpoint of the segment and .Prove that a solution exists if and only if
4． Consider an isosceles triangle.let be the radius of its circumscribed circle and be the radius of its inscribed circle.Prove that the distance between the centers of these two circle is
5． On the circle there are given three distinct points .Construct (using only a straightedge and a compass) a fourth point on such that a circle can be inscribed in the quadrilateral thus obtained.
6． Point and segment are given.Determine the locus of points in space which are vertices of right angles with one side passing through ,and the other side intersecting segment .
7． A circle is inscribed in a triangle with sides .Tangents to the circle parallel to the sides of the triangle are contructe.Each of these tangents cuts off a triagnle from .In each of these triangles,a circle is inscribed.Find the sum of the areas of all four inscribed circles (in terms of ).
8． Consider with acute angle .Thorugh a point perpendiculars are drawn to and ,the feet of which are and respectively.The point of intersection of the altitudes of is .What is the locus of if is permitted to range over
a) the side ;
b) the interior of .
9． Let be a triangle,and let ,,be three points in the interiors of the sides ,,of this triangle.Prove that the area of at least one of the three triangles ,,is less than or equal to one quarter of the area of triangle .
Alternative formulation:Let be a triangle,and let ,,be three points on the segments ,,,respectively.Prove that
,
where the abbreviation denotes the (non-directed) area of an arbitrary triangle .
10． The parallelogram has ,and the triangle has all angles acute.Prove that circles radius and center cover the parallelogram if and only

英语翻译1． Two planes,and ,intersect along the line .The point is given in the plane ,and the point in the plane ; neither of these points lies on the straight line .Construct an isosceles trapezoid (with ) in which a circle can be inscribed,an
1 .,两架飞机沿着这条线相交.问题是在飞机上,在飞机;这些要点在于直线.构建等腰梯形(带),围成一个圆圈,可以刻顶点并躺在飞机和各一份
构建了.2、(三角形、高度和),平均从顶点.
如果一个三角形.构建,哪里是正中段和.证明解的存在,当且仅当
4 .考虑等腰triangle.让半径的圆的半径,受到它的圆.证明之间的距离,这两个圆圈中心
五.圆上有三种不同的分了.构装体(仅使用一个直尺、罗盘)第一点,这样可以在一圈的四边形图像.
6点和段.确定的轨迹,在空间的顶点直角一边穿过,而另一方交叉段.
7 .一圈是在一个三角形.平行切线圆边的三角是contructe.每一种切线triagnle隔断.在每一种三角形,围成一个圆圈是刻.发现的地区的所有4个刻有圈(在).
8.考虑与锐角.送进一分两柱都被吸引到脚,其中,分别.交叉点上的高度.什么是位点被允许范围内的对吗
两边.
b)内部的.
九.我们是一个三角形,让全取三分,将在其内部的两侧,这个三角形.证明面积至少三、三角形、小于或等于四分之一的地区的三角形.
选择配方让一个三角形,让、3分,在部分,分别使用.证明了这一点
(拜托,加多点分嘛,打得很辛苦呢!）