In the figure above, equilateral
triangles ABD and BCE are drawn on the sides of a triangle ABC.
F, G, and H are the midpoints of AD, CE, and AC respectively. HL
and FG are perpendicular. Lines FH, CD, HL, and FG are cut
by line AE at J, O, P, and N respectively. Lines GH, HL, and FG
are cut by line CD at K, Q, and M respectively. Prove the
following:

AE = CD

The measure of angle DOE is 120º

FH = GH

The measure of angle FHG is 120º

mÐHFG = mÐHGF
= 30º

OB is the bisector of ÐDOE

mÐOBC = mÐOEC
= mÐHGC
mÐADO = mÐABO
= mÐAFH

OD is the bisector of ÐAOB
OE is the bisector of ÐBOC

mÐAOB = mÐBOC
= 120º

mÐCMG = mÐANF
= 30º

OB and FG are perpendicular

Triangle OPQ is equilateral

Triangle JPH is equilateral

Triangle HKQ is equilateral

HJ = HK + OP


Using the deductive method, we start
with a few true statements (the axioms) and use them to prove
dozens, hundreds, or thousands, of other statements (the
theorems). Geometry was organized by the Greek mathematician
Euclid, and his deductive method is still used by mathematicians
today.
Hints: See
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