1. Open file:
fermat2.gsp

2. Pictured is the
original construction (A) as well as the modified construction (B) from
the last proof. Circumcircles ADB and BEC are shown where they should
intersect at point B.

3. Now, construct the
other point of intersection of these two circles. This will be your new
point O.

4. Use the segment tool
to construct the six segments OA, OB, OC, OD, OE and OF.

First, we will prove
that AOE and DOC are straight lines, and then that the circumcircle AFC
also passes through O. Using this fact, we will then show that BOF is also
a straight line, which implies that lines AE, DC, and BF are concurrent at
O.

Answer the questions on
a separate piece of paper accordingly to hopefully prove what we need.

*(questions a-j for
proof 2)*

a. What can you say
about the size of angle BCE ? Why?

b. From a, what can
you now say about the size of angle BOE? Why?

c. What can you say
about the size of angle BOA? Why?

d. From b and c, what
can you now conclude about angle AOE?

e. Repeat the same
argument to show that DOC is a straight line.

f. From the angles
determined above, calculate the size of angle AOC.

g. From f, what can
you now conclude about quadrilateral CFAO? Why?

5. Now, we need to
construct circumcircle of triangle AFC. Remember that the center of the
circumcircle is the intersection of the perpendicular bisectors of an
equilateral triangle. Be sure to only show the circle and the center,
hiding the construction lines.

h. Repeat the same
argument as in a- e

i. Would the preceding
argument still be valid if angle ABC were not equal to 90 degrees? What
can you conclude from that?