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Problems Geometry Plane geometry Circles Circles (other)

Two circles intersect at points $A$ and $B$ . Point $X$ lies on the line $A B$ , but not on the segment $A B$ . 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 $∠ C A B = 90^{\circ}$ .

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 $O_{1} A_{1} ∥ O_{2} A_{2}$ .

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 $A B$ and $A C$ are drawn to a circle with center $O$ . Prove that if from the point $M$ the segment $A O$ is visible at an angle of $90^{\circ}$ , then the segments $O B$ and $O C$ 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-56747

Problems Geometry Plane geometry Area Area (other)

13–15 2.0

Let $E$ and $F$ be the midpoints of the sides $B C$ and $A D$ of the parallelogram $A B C D$ . Find the area of the quadrilateral formed by the lines $A E, E D, B F$ and $F C$ , if it is known that the area $A B C D$ is equal to $S$ .

Problem #PRU-56750

Problems Geometry Plane geometry Triangles

12–13 2.0

Let $G, F, H$ and $I$ be the midpoints of the sides $C D, D A, A B, B C$ of the square $A B C D$ , whose area is equal to $S$ . Find the area of the quadrilateral formed by the straight lines $B G, D H, A F, C E$ .

Problem #PRU-56764

Problems Geometry Plane geometry Area Area of quadrilateral

13–14 2.0

The diagonals of the quadrilateral $A B C D$ intersect at the point $O$ . Prove that $S_{A O B} = S_{C O D}$ if and only if $B C ∥ A D$ .

Problem #PRU-56826

Problems Geometry Plane geometry Triangles

12–13 2.0

a) Prove that if in the triangle the median coincides with the height then this triangle is an isosceles triangle.

b) Prove that if in a triangle the bisector coincides with the height then this triangle is an isosceles triangle.