Online Geometry theorems, problems, solutions, and related topics.
Geometry ProblemLevel: Mathematics Education, High School, Honors Geometry, College.Click the figure below to see the complete problem 801.
Let AB=a,CD=b and BD=y1- Δ BCEa²+b²+ab=1002- Δ BCDa²+b²-ab=y²2(a²+b²)=100+y²3- ΔBDC => theorem of apollonius2(a²+b²)=4x²+y²4x²+y²=100+y²4x²=100X=5Erina-NJ
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
Let AB=BC=Ac=a and CD=CE=DE=bApply cosine formula in triangle BCE we have BE^2=a^2+b^2+a.b= 10^2Apply cosine formula in triangle BCD we have BD^2=a^2+b^2-a.bIn triangle BCD apply formula for calculate median we have 2.CF^2=BC^2+CD^2-BD^2/2Or 2.x^2=1/2(a^2+b^2+a.b)= ½.BE^2=50So x=5
1) continue CF until CF=FP => CF = CP/2BCDP is a paralellogram2) E, D, P are collinear. BCEP is an isosceles trapezoid.PC=BE=10 => CF=5Erina-NJ
Problem 801 solution: This solution was submitted by Michael Tsourakakis from Greece Thanks Mixalis.
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.
Complete the parallelogram BCDG.Triangles BCG, BCE are congruent.2x = GC = BE = 10.x = 5
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.
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.
Simple pure geometry solutionDraw FG // to DC where G is on BCConsider Tr.s BCE & FGC< C = < G = 120BC = 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. BCEHence x = 10/2 = 5Sumith PeirisMoratuwaSri Lanka