NONLSSIL ONSTRUTIONS II hristopher Ohrt UL Mthcircle - Nov. 22, 2015 Now we will try ourselves on oncelet-steiner constructions. You cn only use n (unmrked) stright-edge but you cn ssume tht somewhere in the plne there is one circle (nd its center) given. roblem 1 (onstruction of prllels through given point). (i) Just using n (unmrked) strightedge construct the prllel to bisected line segment (tht is line segment of which you re given the midpoint). (ii) Now ssume you re given circle nd its midpoint. onstruct prllelogrm inscribed in the circle. Hint: Strt with rndom dimeter of the circle nd try using the previous result. (iii) You re still given circle nd its midpoint. onstruct bisected line segment on given line (not necessrily pssing through the circle). Hint: hoose rndom dimeter nd intersect it with the line. Now try using wht you ve lerned so fr. (iv) How does this enble you to give the oncelet-steiner construction of the prllel to given line through given point? 1
roblem 2 (Reflection on line). Given point nd line, use the sketch to construct its reflection. The second sketch shows you wht you strt out with. Hint: The dimeter contining H is rndomly chosen. The lines HJ nd re prllel. Wht else is prllel? H J 2
roblem 3 (Intersection of line nd circle). Use the following sketch to construct the intersection points of the circle round O of rdius O nd the line. The second sketch shows wht you strt out with. The dotted circle cnnot be drwn (with stright-edge), so it s imgintively there but you cnnot intersect lines with it directly. Include proof why your construction works. Hint: The point L is freely chosen nywhere on. Things tht look prllel, re most likely prllel. First use ny pprent prellelities nd the intersect theorem to prove your ssertion. Then use the intersect theorem (severl times) to prove tht ll the lines you need re prllel. M L O M O 3
roblem 4. Refer to the ppendix for the definition nd bsic properties of the rdicl xis of two circles. Show tht the (three different) rdicl xis of three circles intersect in one point. n you use this to construct the rdicl xis of two circles using stright edge nd compss? 4
roblem 5 (Rdicl xis of two circles). Given two circles nd b, use the sketch below to construct their rdicl xis. The second sketch shows wht you strt out with. The dotted circles re given (but not drwn). However, thnks to the previous problem you cn merrily intersect ny line with them. Hint: The point L is chosen freely on. So re the rys contining nd B. Show tht BD is concyclic nd conclude from the previous problem tht therefore R must be on the rdicl xis. n you identify second point on the rdicl xis? During your construction, be mindful tht the only thing you know bout the given circles is their midpoint nd rdius. In prticulr you cn NOT drw them/intersect them with other circles. b L B R D b 5
roblem 6. Use the previous two constructions to construct the intersection points of two circles nd thus finish the proof of the oncelet-steiner Theorem. 6
ppendix: Some Fcts from Geometry Recll the following fcts from geometry. If you re curious you cn sk one of the instructors to give you proofs for them. oncyclic oints In the sketch below the points, B,, nd D re on circle if nd only if the ngles BD nd D gree. Similrly, B,, nd D re on circle if nd only if the ngles B nd ED gree. B D E Rdicl xis The rdicl xis of two circles c 1 nd c 2 is the set of points such tht the tngents through to c 1 nd c 2 hve the sme length. Note tht we cn drw two different tngents through to given circle, but they will lwys hve the sme length. It cn be shown tht the rdicl xis is line perpendiculr to the line connecting the centers of c 1 nd c 2. n you describe the rdicl xis of two intersecting circles? The sketch below shows the rdicl xis of two circles nd point on it such tht the tngentil lengths r 1 nd r 2 gree. r 1 r 2 7