## Friday, November 30, 2012

### Problem 829: Triangle, Circumcircle, Parallel, Chord, Side, Metric Relations

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

Click the figure below to see the complete problem 829.

1. x=5sqrt(5/2)

2. Denote BF, BG by p & q resply. Use Stewart's Theorem to obtain: 125+x^2=6(p^2+5), 50+4x^2=6(q^2+8)and p/(p+5/p) =q/(q+8/q).
Solve simultaneously to get x=5√(2.5) --- same as Atharva

3. Since DE//AC so Arc(AD)=Arc(EC) => angle (ABF)=angle(CBG)
So in triangle ABC ,line BF and BG are isogonal to each other.
Per Steiner’s theorem for isogonal lines BF and BG
(AF/FC).(AG/GC)=AB^2/BC^2
So 1/5 . 4/2 =25/x^2
So x=5.sqrt(5/2)

4. Problem 8Since ACED is an isosceles trapezoid, AE = CD = y (say).

Let BF = p, DF = q, BG = u and GE = v

From similar triangles,

∆ABF & ∆DCF ; y/5 = q/1 = 5/p…..(1) and
∆BCG & ∆AEG; y/x = v/2 = 4/u ….(2)

Now since AC//DE ; p/q = u/v and substituting for p, q, u & v from (1) & (2) we have
(25/y)/(y/5) = (4x/y)/(2y/x) or x2 = 125/2 which gives us x = 5√(5/2)

Sumith Peiris
Moratuwa
Sri Lanka