Unizor — Geometry2D — Similarity — Problems 5

1. Given a circle with diameter AB and chord PQ perpendicular to diameter AB and intersecting this diameter at point X.
Prove the following equality between the lengths of corresponding segments:
4·AX·XB = PQ^2.
2. Given a circle with diameter AB and chord PQ intersecting this diameter at point X.
Prove the following equality between the lengths of corresponding segments:
AX·XB = PX·XQ.
3. Given a circle and point X inside it.
Let AB be any chord that contains this point X.
Prove that the product of lengths of segments AX and XB is independent of the position of a chord AB, as long as it contains point X.
4. Given a circle and point X outside it.
Let XM be a tangent with point M being a point of tangency.
Consider any line that contains point X and intersects a circle in two points A and B.
Prove that the product of lengths of segments XA and XB is equal to a square of a distance from X to M:
XA·XB = XM^2
5. Given a circle and point X outside it.
Consider any line that contains point X and intersects a circle in two points A and B.
Prove that the product of lengths of segments XA and XB is independent of the position of this line, as long as it contains point X.

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