An example

Preliminary knowledge: Triangle midsegment theorem (applet by Brzezinski, 2016)

The converse of the intercept theorem, a.k.a. Thales' (first) theorem in many countries, or "Strahlensatz", or "Transversalsatsen", or "Párhuzamos szelők tétele"

If the two intersecting lines are intercepted by two arbitrary lines and a:b=c:d holds then the two intercepting lines are parallel. [br]

Step 1: LocusEquation[a/b==c/d,P]

Step 1 (alternative)

Step 2: Conjecture

It seems that parallelism of e and f implies a/b=c/d.

Step 3: Relation[a/b,c/d]

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Step 4: Find a proof

(Optional.)

Step 5: Let the children play

Some ideas to motivate the children:[list][*]Can e/f be expressed by a and b?[/*][*]How about a·b=c·d?[/*][/list]

Idea 1: LocusEquation[a/(a+b)==e/f,Q]

Idea 2: LocusEquation[a b == c d,P]

Conclusions

[list][*]Why are there two solutions for a:b=c:d? Surprising answer (which may be difficult to interpret), here the harmonic property of the [url=https://en.wikipedia.org/wiki/Cross-ratio]crossed ratio[/url] will play an important role (well known in projective geometry).[/*][*]In the case of the triangle midsegment theorem the harmonic conjugate is at infinity (so there is only one solution).[/*][*]Implicit formulas (due to theoretical difficulties) may introduce unwanted points in the locus output. This is the case for a·b=c·d for the upper oval curve.[/*][*]Factorizing the output formula may enlighten some details. The case for a·b=c·d results in a sextic which resolves into two cubics.[/*][*]Of course for children this is advanced mathematics and will not appear in school. Also for teachers these results may be difficult to interpret. But is this a reason to not support children to just play and discover?[/*][/list]