Problems

Age
Difficulty
Found: 7

The circles σ1 and σ2 intersect at points A and B. At the point A to σ1 and σ2, respectively, the tangents l1 and l2 are drawn. The points T1 and T2 are chosen respectively on the circles σ1 and σ2 so that the angular measures of the arcs T1A and AT2 are equal (the arc value of the circle is considered in the clockwise direction). The tangent t1 at the point T1 to the circle σ1 intersects l2 at the point M1. Similarly, the tangent t2 at the point T2 to the circle σ2 intersects l1 at the point M2. Prove that the midpoints of the segments M1M2 are on the same line, independent of the positions of the points T1,T2.

The point A is fixed on a circle. Find the locus of the point X which divides the chords that end at point A in a 1:2 ratio, starting from the point A.

Two circles touch at point K. The line passing through point K intersects these circles at points A and B. Prove that the tangents to the circles drawn through points A and B are parallel.

Two circles c and d are tangent at point B. Two straight lines intersecting the first circle at points D and E and the second circle at points G and F are drawn through the point B. Prove that ED is parallel to FG.

Prove that the points symmetric to an arbitrary point relative to the midpoints of the sides of a square are vertices of some square.

The points A and B and the line l are given on a plane. On which trajectory does the intersection point of the medians of the triangles ABC move, if the point C moves along the line l?