Tiffany N. KeysŐ Essay 2:
INVESTIGATION: Given any triangle ABC, construct equilateral triangles on each side and find the center of each equilateral triangle. The triangle formed by these three centers is Napoleon's Triangle.
Step 1: Given triangle ABC, construct the point D such that ABD is an equilateral triangle, and D and C are on opposite sides of AB.
Step 2: Construct the point E such that BCE is equilateral.
Step 3: Construct the point F such that CAF is equilateral.
Step 4: Let J be the center of ABD, K the center of BCE, and L the center of CAF.
Step 5: Then JKL is Napoleon's triangle.
Step 1: Construct the point O such that triangles DAO and ABC are congruent.
Step 2: Construct the point N such that BDN and ABC are congruent.
Step 3: Construct the Napoleon triangles JMI and JGH from triangles DAO and BDN.
Step 4: Considering triangle TEB, it is assumed that BE = BC and BT = BN = AC.
Step 5: Since angle DBN is equal to angle BAC, it is assumed that <EBT = 180 - <ABC - <BAC = <ACB.
This shows that triangle TEB must be congruent to triangle ABC by SAS.
Step 6; By construction, GK = KL, which also shows that GK = KL = LM = MI = IH = HG.
Step 7: JL = JG by construction, GK = LK by STEP 6, and JK = JK, so that triangles GJK and LJK are congruent by SSS.
Step 8: Therefore we can assume <KJG = <LJK and that the six angles at J must all be congruent and all are 60ˇ.
Step 9: Since <LJK = 60 degrees, we can assume that <JKL and <KLJ are 60ˇ.
Step 10: In conclusion, triangle JKL is equilateral.