## Friday, June 1, 2012

### Problem 759: Equilateral Triangle, Transversal, Trisection of sides, Congruence, Angle

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the problem 759 details.

1. By Menelaus theorem on triangle ABC, (BD/DC)*(CF/FA)*(AE/EB)=1.
Hence CF/FA = 4/1. Therefore FA:AC=1:3.
Since ABC is an equilateral triangle, FA = (1/3)AC = (1/3)AB = AE
So triangle AEF is an isosceles triangle. Angle EAF = 120, hence x = 30.

2. please disregard my previous solution. new Solution below include minor typo corrections.
Apply Menelaus’s theorem to triangle ABC with secant FED
DB/DC x FC/FA x EA/EB=1 => ½ x FC/FA x ½= 1
So FA/FC= ¼ => CA/CF= ¾
From A draw AH //FD ( H on CB)
We have CA/CF=CH/CD= ¾ => CH=3/4.CD=3/4 .2/3 .CB= ½. CB
AH will be an angle bisector and altitude of triangle ABC, so x= angle(HAC)= 30

3. Draw EG ∥ DC (G on AC)
∆s GDC and AEG are equilateral and
BEGD is a parallelogram
1/2 = EG/DC = FG/FC
G is the midpoint of FC
FG = GC = EB = DG
∆FDC is rt angled at D
x = 30°

4. AE = BD = a/3 where AB = BC = AC = a
So BE = CD = 2a/3
Apply Menelau's Theorem to ∆ABC
and transversal DEF
(BE/EA).(AF/FC).(CD/DB) = 1 numerically
Easy to see BE/EA = 2 = CD/DB
Follows AF/FC = 1/4, AF = a/3 = AE
So x = ∠AFE = ∠AEF = (1/2)∠EAC = 30°

5. Let G be the midpoint of [BE].(Draw a picture yourself)
From BG/BA=BD/BC =1/3 follow that triangles BDG and BAC are similar; moreover DG/CA=1/3.
Now, in triangle BDE, median DG=BE/2- half of side of base, or that suffice that angle BDE is right. In fact, because triangle BDG is equilateral, follow that triangle EBD (in that order of vertices) is of type 30-60-90.
The rest is simple.

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6. Let BD = AE = a . Draw DG // AB intersecting AC at G. now DC = BE = 2a. In tr.ABC, by BPT, GC = 2a and AG = a.
now tr. DGC is equilateral. so DG = 2a . In tr. FGD and tr.FAE are similar. so FA = a. Now FA = AE = a. So ∠FEA = ∠AFE = x. ∠ FEA = ∠ GDF = x. In tr. DFG ∠DFG + ∠ GDF = ∠DGC. 2x = 60° or x = 30°

7. http://www.mathematica.gr/forum/viewtopic.php?f=20&t=27308&p=133642

8. Easily BED is a 30-60-90 Tr. Having BE = 2.BD and the included angle 60. So < BED = 30 and hence x = 30.

Sumith Peiris
Moratuwa
Sri Lanka