Problems

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Two lines CD and CB are tangent to a circle with the center A and radius R, see the picture. The angle BCD equals 120. Find the length of BD in terms of R.

Given two circles, one has centre A and radius r, another has centre C and radius R. Both circles are tangent to a line at the points B and D respectively and the angles CED=AEB=30. Find the length of AC in terms of r and R.

Consider a triangle CDE. The lines CD, DE, and CE are tangent to a circle with centre A at the points F,G, and B respectively. We also have that the angle DCE=120. Prove that the length of the segment AC equals the perimeter of the triangle CDE.

A circle with center A is tangent to the lines CB and CD, see picture. Find the angles of the triangle BCD if BD=BC.

Take two circles with a common centre A. A chord CD of the bigger circle is tangent to the smaller one at the point B. Prove that B is the midpoint of CD.

Prove that the lines tangent to a circle in two opposite points of a diameter are parallel.

CD is a chord of a circle with centre A. The line CD is parallel to the tangent to the circle at the point B. Prove that the triangle BCD is isosceles.

Four lines, intersecting at the point D, are tangent to two circles with a common center A at the points C,F and B,E. Prove that there exists a circle passing through all the points A,B,C,D,E,F.

A circle with center A is inscribed into the triangle CDE, so that all the sides of the triangle are tangent to the circle. We know the lengths of the segments ED=c,CD=a,EC=b. The line CD is tangent to the circle at the point B - find the lengths of segments BD and BC.

A circle with center A is tangent to all the sides of the quadrilateral FGHI at the points B,C,D,E. Prove that FG+HI=GH+FI.