Problems

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Two circles intersect at points A and B. Point X lies on the line AB, but not on the segment AB. Prove that the lengths of all of the tangents drawn from X to the circles are equal.

Let a and b be the lengths of the sides of a right-angled triangle and c the length of its hypotenuse. Prove that:

a) The radius of the inscribed circle of the triangle is (a+bc)/2;

b) The radius of the circle that is tangent to the hypotenuse and the extensions of the sides of the triangle, is equal to (a+b+c)/2.

From a point A the tangents AB and AC are drawn to a circle with center O. Prove that if from the point M the segment AO is visible at an angle of 90, then the segments OB and OC are also visible from it at equal angles.

Two circles have radii R1 and R2, and the distance between their centers is d. Prove that these circles are orthogonal if and only if d2=R12+R22.

Let E and F be the midpoints of the sides BC and AD of the parallelogram ABCD. Find the area of the quadrilateral formed by the lines AE,ED,BF and FC, if it is known that the area ABCD is equal to S.

A polygon is drawn around a circle of radius r. Prove that its area is equal to pr, where p is the semiperimeter of the polygon.

The point E is located inside the parallelogram ABCD. Prove that SABE+SCDE=SBCE+SADE.

The diagonals of the quadrilateral ABCD intersect at the point O. Prove that SAOB=SCOD if and only if BCAD.

Prove that a convex quadrilateral ABCD can be drawn inside a circle if and only if ABC+CDA=180.

Prove that a convex quadrilateral ICEF can contain a circle if and only if IC+EH=CE+IF.