Prove that three common chords of three intersecting circles intersect at one common point?

Question

I hate math formulas and equations and prefer a proof two to three lines of text long.

Answer 1

By suggesting that there ought to be such a short proof for this clues one in on how to do this, because short proofs are very few:

Instead of circles, we have 3 intersecting spheres. The vertical circles of intersection meet at the 2 points common to all three spheres. When viewed from above, one sees how the proof to the question immediately follows.

Answer 2

The equation of a line passes through (0,0) is y = mx................................(i) here c=0 put y=mx in the equation of circle x ^2 + y ^ 2 -3x - 4y -4=0...............(ii) x^ +(mx)^2 -3x - 4(mx) -4 =0 x^(1+m^2) -m(4m+3) -4 = 0 Let x1 and x2 are the roots Therefore (x1+x2) = (4m+3)/(1+m^2)......................(iii... and x1*x2 = -4/(1+m^2)................................. sine the the point(0,0) cuts the it in the ratio of 1:4 (4x1+x2)/ (4+1) =0 4x1 +x2 = 0 x2 = -4x1....................................... From (iv) and (v) - 4x1^2 = -4/(1+m^2) ( x1)^2 = 1/(1+m^2)...........................(vi) Since x1 lies on the Circle, it satisfy it 1/(1+m)^2 +m^2/(1+m^2)