## Saturday, September 1, 2012

### Problem 801: Equilateral Triangles, Cevian, Midpoint, Distance, Measurement, Metric Relations

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

Click the figure below to see the complete problem 801.

1. Let AB=a,CD=b and BD=y
1- Δ BCE
a²+b²+ab=100
2- Δ BCD
a²+b²-ab=y²
2(a²+b²)=100+y²
3- ΔBDC => theorem of apollonius
2(a²+b²)=4x²+y²
4x²+y²=100+y²
4x²=100
X=5

Erina-NJ

2. ED meets AB at H. CH=BD and CF=BG, where F and G are mindpoints of BD and CH respectivly. It is obvious that CF=BG=5.
You can see this here: http://mtz256.files.wordpress.com/2012/09/801.jpg

3. Let AB=BC=Ac=a and CD=CE=DE=b
Apply cosine formula in triangle BCE we have BE^2=a^2+b^2+a.b= 10^2
Apply cosine formula in triangle BCD we have BD^2=a^2+b^2-a.b
In triangle BCD apply formula for calculate median we have 2.CF^2=BC^2+CD^2-BD^2/2
Or 2.x^2=1/2(a^2+b^2+a.b)= ½.BE^2=50
So x=5

4. 1) continue CF until CF=FP => CF = CP/2
BCDP is a paralellogram
2) E, D, P are collinear. BCEP is an isosceles trapezoid.
PC=BE=10 => CF=5

Erina-NJ

5. Problem 801 solution: This solution was submitted by Michael Tsourakakis from Greece

Thanks Mixalis.

6. Sea M el punto medio del lado BC. Entonces el segmento FM es paralelo a DC y mide DC/2, y el triángulo CFM es semejante al triángulo BEC, por el criterio LAL, CM=BC/2, FM=DC/2=EC/2, ángulo FMC = ángulo ECB = 120º, así pues x=CF=BE/2=5.

Migue.

7. Complete the parallelogram BCDG.
Triangles BCG, BCE are congruent.
2x = GC = BE = 10.
x = 5

8. Prof Radu Ion,Sc.Gim.Bozioru,BuzauOctober 29, 2012 at 12:01 PM

Construind simetricul punctului D fata de C si notandu-l cu P obtinem ca
Δ BCE=Δ BCP (LUL) BE=BP si CF linie mijlocie in
Δ BDP <=> x=CF=BE/2=5.

9. BP is parallel to FC such that P is on AC. DC=CP but also DC=EC so CP=CE. <BCE=60+60=120, but also <BCP=180=60=120 so <BCE=BCP. These equalities show that BCE and BCP are congruent, making BC=10=BP=2x so x=5.

10. Simple pure geometry solution

Draw FG // to DC where G is on BC

Consider Tr.s BCE & FGC

< C = < G = 120
BC = 2 GC
CE = 2 FG ( last 2 results from applying mid point theorem to Tr. BCD)

The 2 triangles are therefore obviously similar and the sides of Tr. FGC are half that of Tr. BCE

Hence x = 10/2 = 5

Sumith Peiris
Moratuwa
Sri Lanka