The Equilateral Triangle
Given an equilateral triangle ABC with side length = s.
Find AG = BG = CG.
Find EG = DG = FG.
Find the height of the triangle (h) AD
= BF = EC.
1) AD through G is an angle bisector of <BAC, by construction. AD
is a side bisector of BC.
2) CE through G is an angle bisector of <BCA, by construction. CE is a
side bisector of AB.
3) BF through G is an angle bisector of <ABC, by construction. BF is a
side bisector of AC.
4) triangles AEG, AFG, BEG, CEG, BDG, CDG are all congruent, by side-side side
and by construction.
Find h.
By Pythagorean Theorem (PT),
h² =
h = .
Find the area of ABC.
The area of any triangle = 1/2 * length of base * height.
The base = CA = s.
Area ABC =
Show each of the triangles in 4) is similar to triangles ADC and ADB.
5) DC is common to both triangles DCG and DCA..
6) ADC and CDG are right, and are common to both
triangles DCG and DCA.
7) DCG = DAC because both are bisections of
equal angles BCA and BAC.
8) Triangle DCG is similar to triangle DCA by angle-side-angle:
9) Therefore, CG is to CD as AC is to AD, or,
CG / CD = AC / AD --->
2 * CG = .
CG = AG = BG =
DG = EG = FG =
The longer portion of the diagonal is twice as long as the shorter part:
CG / DG = = 2.