Question about Geometry

[Horse]

Homework Statement

Let ABC be a right angled triangle such that A=90, AB=AC and let M be the mid point of the side AC. Take the point P on the side BC so that AP is vertical to BM. Let H be the intersection of AP ad BM.

Find the ratio BP : PC.

Homework Equations

AB=AC
AM = MC

BP : PC = ?

See the problem here: http://4.bp.blogspot.com/_Qc1Z3hIcYO4/SgbXNoa555I/AAAAAAAABPM/ruupwSz5EoE/s1600-h/bp_pc.JPG.

The Attempt at a Solution

I drew parallel lines, and I found some interesting connections. I am very close to the answer, but I still cannot find it.

Please, see the attachment "parallel_lines.jpg".

 

[Horse]

I will add some of my findings. Please, see the picture!

The Attempt at a Solution

c' = 2b'
c = d + d'

$${AM}^2 = {b}^2 + {{b'}^'}^2$$

where b' = c

As the attachement is pending for Approval, please see the picture about the problem: http://4.bp.blogspot.com/_Qc1Z3hIcYO4/SgbXNoa555I/AAAAAAAABPM/ruupwSz5EoE/s1600-h/bp_pc.JPG.

 

[Horse]

I think I solved it. When I considered my draft at http://4.bp.blogspot.com/_Qc1Z3hIcYO4/SgbXNoa555I/AAAAAAAABPM/ruupwSz5EoE/s1600-h/bp_pc.JPG, I realized that the ratio must be 1:2.

 

[VKint]

Mass points kill this problem in about 2 seconds. Assign mass 2 to points C and A, so point M has mass 4. Using the fact that BAM ~ BFA ~ AFM, you can find that BF/FM = 4, so point B has mass 1. Thus, BP/CP = [ C ]/[ B ] = 2:1 (where [ ] denotes mass).

If you don't know what mass points are, you should go learn about them immediately. (They're awesome.)

 

[Horse]

Mass points kill this problem in about 2 seconds. Assign mass 2 to points C and A, so point M has mass 4. Using the fact that BAM ~ BFA ~ AFM, you can find that BF/FM = 4, so point B has mass 1. Thus, BP/CP = [ C ]/[ B ] = 2:1 (where [ ] denotes mass).

If you don't know what mass points are, you should go learn about them immediately. (They're awesome.)

Great thanks for the awesome tip!