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Perpendicular circles

In document Euclidean plane and its relatives (Page 75-78)

Assume two circles Γ and Ω intersect at two points X and Y. Letℓand

mbe the tangent lines atX to Γ and Ω correspondingly. Analogously,ℓ′

andm′ be the tangent lines atY to Γ and Ω.

We say that the circle Γ isperpendicularto the circle Ω (briefly Γ⊥Ω) if they intersect and the lines tangent to the circle at one point (and therefore, both points) of intersection are perpendicular.

Similarly, we say that the circle Γ is perpendicular to the lineℓ(briefly Γ ⊥ ℓ) if Γ ∩ℓ 6= ∅ and ℓ perpendicular to the tangent lines to Γ at one point (and therefore, both points) of intersection. According to Lemma 5.16, it happens only if the lineℓpasses thru the center of Γ.

Now we can talk about perpendicular circlines.

10.15. Theorem. AssumeΓandΩ are distinct circles. ThenΩ⊥Γ if and only if the circleΓ coincides with its inversion inΩ.

X

Y Q

O

Proof. Let Γ′ denotes the inverse of Γ.

“Only if ” part. LetO be the center of Ω andQ be the center of Γ. LetX and Y denote the points of intersections of Γ and Ω. According to Lemma 5.16, Γ ⊥ Ω if and only if (OX) and (OY) are tangent to Γ.

Note that Γ′ is also tangent to (OX) and (OY)

atX andY correspondingly. It follows thatX andY are the foot points of the center of Γ′ on (OX) and (OY). Therefore, both Γand Γ have

the centerQ. Finally, Γ′ = Γ, since both circles pass thruX. “If ” part. Assume Γ = Γ′.

Since Γ 6= Ω, there is a point P which lies on Γ, but not on Ω. Let

P′ be the inverse of P in Ω. Since Γ = Γ, we have that P Γ. In

particular, the half-line [OP) intersects Γ at two points. By Exercise 5.12,O lies outside of Γ.

As Γ has points inside and outside of Ω, the circles Γ and Ω intersect. The latter follows from Exercise 3.20.

LetX be a point of their intersection. We need to show that (OX) is tangent to Γ; that is,X is the only intersection point of (OX) and Γ.

AssumeZis another point of intersection. SinceOis outside of Γ, the pointZ lies on the half-line [OX).

LetZ′denotes the inverse ofZ in Ω. Clearly, the three pointsZ, Z, X

lie on Γ and (OX). The latter contradicts Lemma 5.14.

It is convenient to define the inversion in the line ℓ as the reflection inℓ. This way we can talk aboutinversion in an arbitrary circline. 10.16. Corollary. Let Ω and Γ be distinct circlines in the inversive plane. Then the inversion inΩsendsΓ to itself if and only if Ω⊥Γ. Proof. By Theorem 10.15, it is sufficient to consider the case when Ω or Γ is a line.

Assume Ω is a line, so the inversion in Ω is a reflection. In this case the statement follows from Corollary 5.7.

If Γ is a line, then the statement follows from Theorem 10.11. 10.17. Corollary. Let P andP′ be two distinct points such that Pis the inverse of P in the circle Ω. Assume that the circline Γ passes thru

P andP′. Then Γ.

Proof. Without loss of generality, we may assume thatP is inside andP′

is outside Ω. By Theorem 3.17, Γ intersects Ω. LetA denotes a point of intersection.

Let Γ′ denotes the inverse of Γ. Since A is a self-inverse, the points

A, P and P′ lie on Γ. By Exercise 8.2, Γ= Γ and by Theorem 10.15,

Γ⊥Ω.

10.18. Corollary. Let P and Qbe two distinct points inside the circle

Ω. Then there is a unique circline Γ perpendicular to Ω, which passes thruP andQ.

Proof. LetP′ be the inverse of the pointP in the circle Ω. According to

Corollary 10.17, the circline is passing thruP andQis perpendicular to Ω if and only if it passes thru P′.

Note thatP′ lies outside of Ω. Therefore, the points P,PandQare

distinct.

According to Exercise 8.2, there is a unique circline passing thru P,

QandP′. Hence the result.

10.19. Exercise. LetΩ1 andΩ2be two distinct circles in the Euclidean

plane. Assume that the point P does not lie on Ω1 nor on Ω2. Show

that there is a unique circline passing thru P which is perpendicular to

Ω1 andΩ2.

10.20. Exercise. LetP,Q,P′ andQbe points in the Euclidean plane. Assume P′ and Qare inverses of P and Qcorrespondingly. Show that the quadrilateral P QP′Qis inscribed.

10.21. Exercise. Let Ω1 andΩ2 be two perpendicular circles with cen-

ters at O1 and O2 correspondingly. Show that the inverse of O1 in Ω2

coincides with the inverse of O2 inΩ1.

10.22. Exercise. Three distinct circles — Ω1,Ω2 andΩ3, intersect at

two points — A and B. Assume that a circle Γ is perpendicular to Ω1

10.23. Exercise. Assume you have two construction tools: thecircum- toolwhich constructs a circline thru three given points, and a tool which constructs an inverse of a given point in a given circle.

Assume that a point P does not lie on the two circles Ω1, Ω2. Using

only the two given tools, construct a circline Γ that passes thru P, and perpendicular to bothΩ1 andΩ2.

In document Euclidean plane and its relatives (Page 75-78)