please help solve: isosceles triangle ABC, right angled at B, w/ square inscribed

[sukumar460]

an isosceles triangle ABC, right angled at B, has a square inscribed inside it, with 3 vertices of the square on 3 sides of the triangle, touching AB at x, BC aty and AC at z. find the ratio between Bx and By.

 

[Deleted member 4993]

an isosceles triangle ABC, right angled at B, has a square inscribed inside it, with 3 vertices of the square on 3 sides of the triangle, touching AB at x, BC aty and AC at z. find the ratio between Bx and By.

What are your thoughts?

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[sukumar460]

Yes am stuck at beginning. Please help me out how to proceed.

 

[Deleted member 4993]

Yes am stuck at beginning. Please help me out how to proceed.

What is the measure of the angle BAC? angle BCA?

Do you know Pythagoras's Theorem?

Do you know laws of sines and cosines for a triangle?

 

[sukumar460]

Ya i know. Angle BAC and angle BCA both will be 45 degrees as it is an isosceles right angled triangle.
I also proceeded with drawing a diagonal YZ, that made another right angled triangle XYZ, right angled at X inscribed in triangle ABC.
BUT how do i proceed with finding the lenght of BY and BX.

 

[Deleted member 4993]

Ya i know. Angle BAC and angle BCA both will be 45 degrees as it is an isosceles right angled triangle.
I also proceeded with drawing a diagonal YZ, that made another right angled triangle XYZ, right angled at X inscribed in triangle ABC.
BUT how do i proceed with finding the lenght of BY and BX.

Do you know laws of sines and cosines for a triangle?

 

[Bob Brown MSEE]

Pick an easy case

Pick an easy case and save a lot of work, and use no Trig. Work backward from guess that Bx/By = 2 (it is good guess by GeoGebra).


I love this sort of problem. You are given that there is a ratio. Clearly it is independent of length AB by scaling symmetry. Pick an especially convenient case. Let AB=10 and point B is origin.


Choose these ...
B=(0,0), A=(0,10), C=(10,0)


Guess that
y=(1,0), x=(0,2) and z=(2,3)
Notice vector xy is -1 across and 2 up.
Notice vector zx is 2 across and 1 up.
Proving that zxy is a right angle.


Prove that
z lies on line AC
done