Two Parallel Lines And A Diverging Line

Number of solutions: 2

SOLUTION METHOD

Translation

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Steps

1. We find the center axis between the parallel lines. The centers of the circles from the solution have to lie on this line.
2. We draw a line perpendicular to the two parallel lines from the assignment.
3. We draw a circle that has its centre at the intersection of the perpendicular and the axis and the point of contact with the parallel lines from the assignment is at the intersection with the perpendicular line.
4. We draw a line perpendicular to the intersecting line from the assignment.
5. We draw a line parallel to the one from 4), which passes through the center of drawn circle.
6. We draw lines that are parallel to the parallel lines from the assignment and also pass through the intersection of the circle and the line from 5).
7. We draw lines perpendicular to the parallel line from the assignment that pass through the intersection of the parallel line from the problem and the lines from 6).
8. We draw circles that are centred on the intersection of the axis from 1) and the lines from 7).

SOLUTION METHOD

Loci of points satisfying certain properties

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Steps

0. Three straight lines are given, two of which are parallel.
1. The centers of the circles have the same distance to all of straight lines. Such points lie on the axes of the angles given by the specified lines.
2. The centers of the circles lie at the intersections of these axes. At the same time, they must lie on the axis of the belt between the specified parallels.
3. The radius of the solution circles is given by the perpendicular distance of the centers from the specified straight lines.
4. The problem has two solutions.

SOLUTION METHOD

Homothety

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Steps

0. Three straight lines are given, two of which are parallel.
1. We will use homothety in the solution. First, we draw a circle that touches the two given straight lines. We choose its radius. At the same time, we draw a tangent to this circle, which is parallel to the line p₂.
2. The drawn circle and the line are the images of the circle of the solution and the line p₁ in homotethy with centre point C.
3. The tangent point T₁ is displayed to the point T'. Therefore, both points must lie on the same line with point C.
4. For the same reason, the center of the solution S₁ lies on the line CS'. At the same time, it lies on the perpendicular line to the line p₁ led by the tangent point T₁.
5. The first solution has its center at point S₁ and passes through point T₁.
6. We draw a parallel to the line p₃, which is also a tangent to the circle k'. Now we will look at the circle k' and the two parallel tangents as the image of the second solution we are looking for.
7. In the second homothety, the intersection of the lines p₁ and p₃ is mapped to the intersection of the tangents f and g. The homothety center lies on the connecting line of these intersections and on the self-joining line p₂.
8. The point T' is the image of the tangent point T₂.
9. The center S' is the image of the point S₂ which is the centre of the solution. At the same time, the center S₂ lies on the perpendicular to the line p₁ passing through the point T₂.
10. The second solution has its center at point S₂ and passes through point T₂.
11. The task has two solutions.