Interesting Locus Result

Investigation
Consider a straight line 𝑓 and a fixed point A not on 𝑓. Create a series of straight-line segments by joining point A to a variety of different points on 𝑓 as illustrated in the dynamic sketch below. Next construct the midpoint of each of these line segments.
1) Click on the 'Show line' button. What do you notice?
2) Drag point B to trace out its locus as it moves along line 𝑓 to check your observation in 1).
3) Click on the 'Hide line' button & drag point A to a different position.
4) Repeat steps 1) and 2). What do you notice?

Formulate
5) Formulate a conjecture of your observations in 1) and 4).

Interesting Locus Result

Conjecture
You should've noticed that the locus of the point B is a straight line parallel to line 𝑓 (and passes through the midpoints of any other line segments drawn from A to 𝑓).

Challenge
6) Can you explain why (prove that) your conjecture is true? Can you prove it in more than one way?
Hint: Try using this well-known theorem: Triangle Midpoints.
7) Can you also prove your conjecture analytically (algebraically)?
8) Compare your solution to 7) to Samson & Stupel (2025).

Note
The above locus result for a straight line can be viewed as an extension of the Triangle Midpoint Theorem.

Further Generalization
If this works for straight lines, would it also work for other functions?
7) Click on the 'Link to parabola' button to navigate to a new sketch.
8) Drag point B to trace out its locus as it moves along the parabola. What you notice?
9) Drag point A to a different position and repeat step 8). What do you notice?
10) Formulate a conjecture.

Challenge 2
11) Can you explain/prove your conjecture in 10? Can you prove it in more than one way?
Hints: Try proving it analytically/algebraically, or geometrically, using this general result from transformation geometry: Mystery Transformation. 12) From the Mystery Transformation above, formulate a general result for any geometric curve.

Reference
Samson, D. & Stupel, M. (2025). An Interesting Locus Result. Learning & Teaching Mathematics, no. 39, pp. 22-25.

External Links
Midpoint theorem (triangle) (Wikipedia)
Dilation (geometry) (Wikipedia)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Michael de Villiers, created with WebSketchpad, 28 Jan 2026.