Relative Positions Of A Line And A Circle / Discussion

This is the discussion related to the wiki page Relative Positions Of A Line And A Circle.

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Exercise 239. What is required to be shown?

Nain415 18 Jul 2018 20:02

Problem: Find the geometric locus of points from which the tangents drawn to a given circle are congruent to a given segment.

Do I understand this correctly? Let the given segment be 'a'. If I choose a point P in the plane, and draw two tangents to some circle, and mark the tangency points A and B, then if PA = PB = 'a', the point P satisfies the condition for membership of the locus of points?

If that's the case, then the problem is quite easy once one learns the solution for exercise 249.

Writing this helped me understand the nature of the problem.

I've completed all of the problems in this chapter except for the very first. Once I have some time (probably 2-3 weeks from now), I'll dedicate some time to posting solutions to exercises in chapters 2.1 and 2.2. So far, I've had a very great time learning elementary geometry with this book. If a solution set to all (or nearly all) problems existed, then I think that this would be more readily adopted by more educators, and therefore more students would receive a rich elementary geometry experience.

Last edited on 18 Jul 2018 20:06 by Nain415 Show more

Charles (guest) 06 Nov 2021 18:55

Page 90, line 4-5: (slants are grater than the perpendicular)

Charles (guest) 06 Nov 2021 19:09

Page 91, figure 123, there seems to be a small gap between EDF and the circle despite EDF being the tangent.

Charles (guest) 06 Nov 2021 20:55

239. Find the geometric locus of points from which the tangents drawn to a given circle are congruent to a given segment.

The geometric locus is a donut with inner diameter just less than the diameter of the given circle and outer diameter congruent to the hypotenuse of a right triangle with legs congruent to the radius of the given circle and the given segment.

Charles (guest) 06 Nov 2021 21:00

240. Find the geometric locus of centers of circles described by a given radius and tangent to a given line.

I'm assuming tangent to a given line should mean tangent to the circles?

Then the geometric locus would be the two parallel and congruent lines to the given tangent a distance of the given radius away, like all the lines are all lined up like ≡.

Charles (guest) 07 Nov 2021 00:55

oh, the circles are tangent to the given line

Charles (guest) 06 Nov 2021 21:59

241. Two lines passing through a point M are tangent to a circle at the points A and B. The radius OB is extended past B by the segment BC=OB. Prove that ∠AMC=3∠BMC.

Notice ∠AMC=∠AMO+∠OMB+∠BMC. Since BM is tangent at B and AM is tangent at A, using 113(2), ∠OBM=d and ∠OAM=d. Since BC is the extension of OB, ∠OBM+∠MBC=2d, so ∠MBC=d. Thus, since BC=OB, ∠OBM=∠MBC, and BM is a common side, △OMB=△BMC by SAS which implies ∠BMC=∠OMB. Since OB and OA are both radii of the same circle, they're congruent. Then, using 55(2), △OMB=△AMO because they're right triangles that share a hypotenuse and have congruent legs OB and OA, which implies ∠OMB=∠AMO=∠BMC. Thus, ∠AMC=3BMC.

Charles (guest) 07 Nov 2021 01:06

242. Two lines passing through a point M are tangent to a circle at the points A and B. Through a point C taken on the smaller of the arcs AB, a third tangent is drawn up to its intersection points D and E with MA and MB respectively. Prove that (1) the perimeter of △DME, and (2) the angle DOE (where O is the center of the circle) do not depend on the position of the point C.

Using 55(2), △OCD=△OAD and △OCE=△OBE because they share hypotenuses and have a common leg congruent the radius, which implies AD=DC, CE=EB, ∠AOD=∠DOC, and ∠COE=∠EOB. Notice, DE=DC+CE, so DE=AD+EB, but also AD+MD=MA and EB+ME=MB, so the perimeter DE+MD+ME=MA+MB. Now, all four angles make up ∠AOB, so 2∠DOC+2∠COE=2∠DOE=∠AOB so ∠DOE=½∠AOB. Both MA+MB and ½∠AOB do not depend on the position of C.

Charles (guest) 07 Nov 2021 01:08

243. On a given line, find a point closest to a given circle.

Drop the perpendicular from the center of the given circle to the given line; the foot should be the closest point.

Charles (guest) 07 Nov 2021 01:11

244. Construct a circle which has a given radius and is tangent to a given line at a given point.

Erect a perpendicular at the given point, and from the given point mark a segment congruent to the given radius. Construct a circle centered on that mark with radius congruent to the given radius.

Charles (guest) 07 Nov 2021 01:16

245. Through a given point, draw a circle tangent to a given line at at another given point.

Connect the two given points and use §67 to construct its perpendicular bisector. Then erect the perpendicular at the given tangency point (the given point on the given line) towards the perpendicular bisector. Construct the circle at the intersection of these two perpendiculars with radius congruent to the line from the intersection to either given point.

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