Kasey Nored

Our problem is given points A, B, and C, construct points X and Y such that AX = XY = YB

Above are A, B, and C, our given points. Lets pick an arbitrary point on AC, say D.

Then construct a circle with a center at D and a radius of segment DA.

Construct a line through point D, parallel to segment CB.

Now let's label some intersections. E, as the intersection of our constructed parallel line and our circle D. F as the intersection of segment CB and our circle D.

Construct segment DF and a line parallel to DF through E. Label the intersection of our newly constructed parallel and segment CB as G. We have now constructed rhombus DEGF.

Constructing circles at F and G with radii of DF shows a nice visual that segments DE, EG, GF, and FD are congruent.

We can translate our constructed Rhombus DEGF to meet the conditions set in the problem.

This is a bit messy, if we hide our construction of DEGF, we can see that XY meet the conditions of the problem. Construction of Circle Y with a radius of AX shows that AX = XY = YB.

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