Question A: Is the statement, three distinct noncollinear points determine a uni
ID: 2982702 • Letter: Q
Question
Question A:
Is the statement, three distinct noncollinear points determine a unique circle, a theorem in Neutral Geometry? Is it also a theorem in Hyperbolic Geometry? Explain.
Answer A:
I think that both answers are No but I can't figure out how to explain why.
Question B:
The two circles shown are internally tangent at point A. The center of the larger circle is B, while the center of the smaller circle is G. The length of CD is 90mm, while the length of EF is 50mm. AD is perpendicular to FB. Determine the length of the diameter of each circle.
Question C:
Two circles are said to be orthogonal if the radius drawn from one of the circles to a point of intersection is perpendicular, at the point, to the radius drawn from the other circle. Prove that if two orthogonal circles have two points of intersection, the radii are perpendicular at both points of intersection.
Explanation / Answer
A)
It is in Euclidean Geometry
And both thementioned geometries are Non- Euclidean
B)
Please post the figure
C)
Define the line segment connecting the two centers of the circles to be line segment AB, where A is the center of one circle and B is the center of another. Let P be the point of intersection given. Now extend AB to a line. Reflect all points on both circle over this line and consider these circles separately. As all circles possess a line of symmetry down the center, the new figure defined by circles A' and B' must be congruent to the initial figure. P' cannot have the same orientation relative to A'B' as P because we are given that there are two points of intersection, and P' lies <= the line (in y) and P lies >= the line (in y). Hence, we can label P'' at the same position as P on the new circle P'. However, as reflection preserves angles, we have that A'P'B' is a right angle, just as A'P''B' is a right angle. As the given figure is congruent to the initial circle, the initial circle must also possess this property.