The problem involves two chords of a circle, AB and CD, intersecting at point X, with given lengths for segments AX, BX, CX, and DX. Participants are tasked with calculating the radius of the circle, which is stated to have an integer value.
The discussion is ongoing, with participants exploring different interpretations of the problem setup. Some suggest that there may not be enough information to definitively determine the radius, while others propose potential reasoning based on the relationships between the segments. A few participants express uncertainty about the assumptions made regarding the chords and their configurations.
There is a mention of a specific section in a book related to intersecting chords and secant theorems, which may impose certain constraints or expectations on the problem-solving approach. Additionally, the original poster notes the need for further information regarding the lengths of certain segments to progress.
StatusX said:I don't think there's enough information here. Take any circle of sufficiently large size (this will be clear in a minute). Draw a chord of length 12, and call the endpoints A and B. Then draw a line from the center of this chord X, to a point C on on the arc between A and B, such that the length of CX is 4. Note this is possible since the maximum length such a segment could have is clearly 6, and the minimum can be made arbitrarily small by increasing the radius of the circle. Now extend this line to intersect the circle again at a point D, and since AX*BX=CX*DX, we must have DX=9. Am I misssing something, or is this exactly the setup you've described? Maybe you're asked to find the smallest such circle?