We solve problems

Problems Geometry Plane geometry Circles Circles (other)

Prove that from the point \(C\) lying outside of the circle we can draw exactly two tangents to the circle and the lengths of these tangents (that is, the distance from \(C\) to the points of tangency) are equal.

Problem #PRU-56654

Problems Geometry Plane geometry Circles Circles (other)

12–14 2.0

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.

Problem #PRU-56656

Problems Geometry Plane geometry Circles Circles (other)

12–14 3.0

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 \((a + b - c)/2\);

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

Problem #PRU-56672

Problems Geometry Plane geometry Circles Tangent lines and tangent circles Tangent circles

13–13 2.0

Two circles touch at a point \(A\). A common (outer) tangent touching the circles at points \(C\) and \(B\) is drawn. Prove that \(\angle CAB = 90^{\circ}\).

Problem #PRU-56673

Problems Geometry Plane geometry Circles Tangent lines and tangent circles Tangent circles

13–13 2.0

Two circles \(S_1\) and \(S_2\) with centers \(O_1\) and \(O_2\) touch at the point \(A\). A straight line intersects \(S_1\) at \(A_1\) and \(S_2\) at the point \(A_2\). Prove that \(O_1A_1 \parallel O_2A_2\).

Problem #PRU-56684

Problems Geometry Plane geometry Circles Tangent lines and tangent circles Tangent lines to circles

13–14 2.0

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 \(90^{\circ}\), then the segments \(OB\) and \(OC\) are also visible from it at equal angles.

Problem #PRU-56707

Problems Geometry Plane geometry Circles Circles (other)

13–14 2.0

Two circles have radii \(R_1\) and \(R_2\), and the distance between their centers is \(d\). Prove that these circles are orthogonal if and only if \(d^2 = R_1^2 + R_2^2\).

Problem #PRU-58084

Problems Geometry Plane geometry Circles Circle, sector, segment, etc Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Pigeonhole principle (finite number of poits, lines etc.)

12–13 4.0

There are 25 points on a plane, and among any three of them there can be found two points with a distance between them of less than 1. Prove that there is a circle of radius 1 containing at least 13 of these points.

Problem #PRU-58094

Problems Geometry Plane geometry Circles Central angle. Arc length and circumference Connection between the arc length, the angle, and the chord Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 4.0

Several chords are drawn through a unit circle. Prove that if each diameter intersects with no more than \(k\) chords, then the total length of all the chords is less than \(\pi k\).

Problem #PRU-58097

Problems Geometry Plane geometry Circles Central angle. Arc length and circumference Plane transformations Projection onto a line Orthogonal projection Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–16 3.0

Several circles, whose total length of circumferences is 10, are placed inside a square of side 1. Prove that there will always be some straight line that crosses at least four of the circles.