Find the Intersection of Two Straight Lines

Geometric Construction with the Compass Alone

Find the intersection point of two lines each given by a pair of points - AB and CD, respectively.

Solution

The solution takes several steps:

1. Construct C₁ and D₁ symmetric to C and D with respect to AB (Problem #2). The required point X can be found in the intersection of CD and C₁D₁.
2. Complete the parallelogram CC₁D₁E (Problem #6).
3. Find x from DE/DD₁ = CD/x (Problem #10).
4. Draw two circles of radius x centered at D and D₁. The required point X lies in the intersection of the two circles.

Problems (Use a compass only)

1. Multiply a Line Segment by a Whole Number
2. Reflect a Point in a Line Sgement
3. Cross a Circle by a Line Segment
4. Drop a Perpendicular to a Line from a Point
5. Detect Collinearity
6. Complete a Parallelogram
7. Bisect an Arc
8. Find the Points of Intersection of a Circle with a Line Segment
9. Build a Square on a Given Side
10. Find the Fourth Proportional of Three Lengths
11. Find the Intersection of Two Straight Lines
12. Divide a Line Segment into a Whole Number of Parts
13. Find the Center of a Given Circle
14. Bisect a given line
15. Mascheroni Construction of a Regular Pentagon

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