Is Point M on the Circumscribed Circle of Square ABCD?

[veljko_c]

Let on side BC of square ABCD set point E, BE:EC=1:2. On side DC point F
(D-C-F), CF:DF=1:3.
Prove that point M = intersection AE and BF, is on circle about ABCD.

Please help.
Thnx

 

[HallsofIvy]

Let on side BC of square ABCD set point E, BE:EC=1:2. On side DC point F
(D-C-F), CF:DF=1:3.
Prove that point M = intersection AE and BF, is on circle about ABCD.

Please help.
Thnx

Here's one way, not necessarly the best but just because I tend to be "analytic". Set up a coordinate system in which the origin is at the center of the square, side AB is x= -1, side CD is x= 1, side BC is y= -1, and side AD is y= -1.

B is (-1, -1) and C is (1, -1) so E is (x,-1) with 1- x= 2(x- (-1)) so 1- x= 2x+ 2 or 3x= -1 so x= -1/3. E is (-1/3, -1) and the line through A and E, the line through (-1, 1) and (-1/3,-1), is y= -3(x+1)+ 1= -3x- 2.

D is (1, 1) so F is (1, y) with y+1= 3(y- 1)= 3(y- 1) so y+ 1= 3y- 3 or 2y= 4 so y= 2. F is (1, 2) and the line through B and F, the line through (-1, -1) and (1, 2) is y= (3/2)(x+1)-1= (3/2)x+ 1/2.

M is the point at which those two lines intersect so you can find it by solving the two equation simultaneously. Then find the distance from that point to the center of the square, (0,0), and show that is the same as the distance from (0,0) to each of the vertices of the square.

 

[tiny-tim]

Welcome to PF!

Hi veljko_c! Welcome to PF!

Let on side BC of square ABCD set point E, BE:EC=1:2. On side DC point F
(D-C-F), CF:DF=1:3.
Prove that point M = intersection AE and BF, is on circle about ABCD.

I'm confused … the way I've drawn it, they intersect inside the square.

 

[chicopee]

I am a little bit "befuddled" about the sentence "point on a circle about square", however I am assuming that you are referring about four corners of a square on a circle . The proper sentence should be " circumscribing the square on the circle" that way there is no confusion.

Nonetheless, letting AB=BC=CD=DA =a; E=a/3 from point B; F=a/4 from point C. Point M, by visual examination, is inside the square and the circle and is not at the center of the circle.

 

[tiny-tim]

Prove that point M = intersection AE and BF, is on circle about ABCD.

I'm confused … the way I've drawn it, they intersect inside the square.

Oh I get it now …

M is on the circle that fits inside the square …​

so define x and y coordinates for A B C and D, find the gradients of AE and BF, and write an equation y = (gradient)x + constant for both, and eliminate y, and show that x² + y² = AB²/4

 

[chicopee]

Tiny Tim, are you saying the the square inscribes the circle, another word, the sides of the square are tangent to the circle?

 

[chicopee]

I just think that the original question was badly worded.
Good going on the solution Tiny-Tim

 

[belliott4488]

We need the OP to clarify the problem, but I took the parenthetical expression "(D-C-F)" to mean that the point F is on the extended line CF, i.e. it is not on the side of the square but is outside of it. If you draw it this way, then the two lines intersect outside the square.

I believe this is how HallsOfIvy interpreted it as well, although I would have put point F at (1,-2), taking the ordering D-C-F to mean that F must be below the point C.