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:

1. AE = CD

2. The measure of angle DOE is 120º

3. FH = GH

4. The measure of angle FHG is 120º

5. mÐHFG = mÐHGF = 30º

6. OB is the bisector of ÐDOE

7. mÐOBC = mÐOEC = mÐHGC
   mÐADO = mÐABO = mÐAFH

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

9. mÐAOB = mÐBOC = 120º

10. mÐCMG = mÐANF = 30º

11. OB and FG are perpendicular

12. Triangle OPQ is equilateral

13. Triangle JPH is equilateral

14. Triangle HKQ is equilateral

15. HJ = HK + OP

16. 

17. 

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.

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