Inversion Problems PDF

Preview

Summary

Questions and solutions ...

Description

Inversion Problems for February 12, 2009 Problem 1. Given a circle with centre O, diameter AB, and any third point C on the circumference, prove that the circle AOC is perpendicular to the circle BOC . Solution. Let O(r) be any circle centred at O with r != 12 |AB |. The mapping σO(r) maps points A, B, C into the points A∗ , B ∗ , C ∗ . Since the line AB passes through O, the line A∗ B ∗ is the line AB . The circle ABC , because it does not pass through O, is mapped into another circle A∗ B ∗ C ∗ . But note that because line AB is a diameter of the circle ABC , they are perpendicular. Since inversion preserves angles, line A∗ B ∗ is also perpendicular to the circle A∗ B ∗ C ∗ , and so is a diameter. Circle AOC maps into line A∗ C ∗ and circle BOC maps into line B ∗ C ∗ . Thus we see that the angle A∗ C ∗ B ∗ is the angle subtended at the circumference of the circle A∗ B ∗ C ∗ by the diameter A∗ B ∗ , and so is a right angle. Since inversion preserves angles, the angle between circles AOC and BOC is also a right angle. Note that in Figure 12 we have drawn the given figure and the inverted figure separately.

C C*

A

O

B

A*

!

O(r)

B*

O

Fig. 12

Problem 2. Let two orthogonal circles intersect in A and B. For any point C on the first circle and any point D on the second circle, prove that the circle through A, C, and D is perpendicular to the circle through B, C, and D . A

D* !

D

A(r)

C

B*

B

Fig. 13

1

C*

Problem 3. If A, B, C, D are not concyclic (i.e. do not lie on the circumference of a circle) and no three of them collinear, prove that the circles through A, B, C and A, B, D intersect in the same angle as the circles through A, C, D and B, C, D. Problem 4. Let A, B, C, D be four concyclic points, i.e. they lie on the circumference of a circle C1 . Let C2 be any circle through A and B and let C3 be any circle through C and D, and let C2 and C3 be tangent in the point Ti . Prove that the locus of these point of contact Ti is a circle. Problem 5. If P Q, RS are common tangents to two circles P AR, QAS, show that the two circles P AQ, RAS are tangent to each other. Problem 6. Let A, B be two fixed points on a given line l, let O be a fixed point not on l, and let X be a variable point on l. Prove that the angle between the circles AOX and BOX is independent of the position of X on l. Problem 7. Let AC be a diameter of a given circle, and let the chords AB, CD intersect in a point O, inside or outside the circle. Show that the circle OBD is orthogonal to the given circle.

2...