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Pentagon

Find the vertices of a pentagon given the midpoints of its sides.

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Triangle Mid Points

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

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Pareq Exists

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

Two Points Plus One Line

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
Simple hint: how would you find the set of points that are equidistant from two given points?

When might that locus not intersect with the given line?

Harder, last part: try dropping perpendiculars from $A$ and $B$ onto the given line.

The distance between those positions might be useful.

When is $P$ between those positions and when is it outside?