I am studying, and there are just 3 problems I don't quite understand. Plz help me.
1. Let Γ be a circle with center Q, let [AB] and [CD] be chords of Γ(so that the endpoints lie on the circle), and let G and H be the midpoints of [AB] and [CD]. Prove that d(Q, G) = d(Q, H) if and only if d(A, B) = d(C, D), and d(Q, G) < d(Q,H) if and only if d(A, B) > d(C, D).
2. Let Γ be a circle with center Q, and let L be a line containing a point X on
Γ. Prove that X is the only common point of Γand L if and only if QX is perpendicular to L. (These are the usual synthetic descriptions for the tangent line to Γat X.) [ Hint : If L also meets Γat another point Y, explain why angleQXY is acute. ]
3. Let Γ be a circle with center Q, let X be a point in the exterior of Γ, and let A and B be two points of Γwhich lie on opposite sides of QX such that XA and XB are tangent to Γin the sense of the preceding exercise. Prove that d(X, A) = d(X, B).
L is tangent to
the line connecting X and the center of is perpendicular to L..
if we use the hint..
suppose L meets at points X and Y.. then is an isosceles triangle with . hence,
now, if , then the sum of the internal angles of would be >
therefore, .. QED
Here's #3 . . .
3. Let C be a circle with center Q.
Let X be a point in the exterior of C,
and let A and B be two points of C which lie on opposite sides of QX
such that XA and XB are tangent to C.
Prove that: .Code:* * * * * A * * * / * * / * * / * * * Q* - - - - * - - - *X * \ * * \ * * \ * * * * * * B * * *
Tangent XA is perpendicular to radius QA.
Tangent XB is perpendicular to radius QB.
. . Hence, triangles QAX and QBX are right triangles.
Since QA and QB are radii of circle Q,
. . And, of course: .