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 \(FG+HI = GH+FI\).
Two circles with centres \(A\) and \(C\) are tangent to each other at the point \(B\). Both circles are tangent to the sides of an angle with vertex \(D\). It is known that the angle \(\angle EDF = 60^{\circ}\) and the radius of the smaller circle \(AF=5\). Find the radius of the large circle.
Two circles with centres \(A\) and \(C\) are tangent to each other at the point \(B\). Two points \(D\) and \(E\) are chosen on the circles in such a way that a segment \(DE\) passes through the point \(B\). Prove that the tangent line to one circle at the point \(D\) is parallel to the tangent line to the other circle at the point \(E\).
Recall that a line is tangent to a circle if they have only one point of intersection, a circle is called inscribed in a polygon if it is tangent to every side as a segment of that polygon.
In the triangle \(CDE\) the angle \(\angle CDE = 90^{\circ}\) and the line \(DH\) is the median. A circle with center \(A\) is inscribed in the triangle \(CDH\) and is tangent to the segment \(DH\) in its middle, let’s denote it as \(G\), so \(GH=DG\). Find the angles of the triangle \(CDE\).
A circle with center \(A\) is inscribed into a square \(CDFE\). A line \(GH\) intersects the sides \(CD\) and \(CE\) of the square and is tangent to the circle at the point \(I\). Find the perimeter of the triangle \(CHG\) (the sum of lengths of all the sides) if the side of the square is \(10\)cm.
Recall that a line is tangent to a circle if they have only one point of intersection, a circle is called inscribed in a polygon if it is tangent to every side as a segment of that polygon.
In the triangle \(EFG\) the line \(EH\) is the median. Two circles with centres \(A\) and \(C\) are inscribed into triangles \(EFH\) and \(EGH\) respectively, they are tangent to the median \(EH\) at the points \(B\) and \(D\). Find the length of \(BD\) if \(EF-EG=2\).