Angle DFA is also a right angle (by construction). Note that angle ADC is a right angle (since it is an inscribed angle to the diameter of circle C). Your resulting figure should look similar to the diagram below. Mark the intersection of this line with AB and call it point F. Claim: AE = 1/3 AB Note: For this construction, you will also need to construct a line perpendicular to AB through point D. Mark the point where circle D intersects segment AB and call it E. Construct a circle with center D and radius AB. Mark the point where circle C and circle A intersect call it D. Construct a circle with center C and radius AC. Mark the point where circle B intersects line AB and call it C. Then, take the diameter of circle A and construct a circle with center B and radius of this diameter length. Construct a circle with center A and radius AB. To begin this construction, start with line segment AB. Trisection Construction #4: For a GSP script tool of this construction, click here. Side AF must be similar to side HD, and since HD = 1, AF = 1/3. Since AE = 1, AND EH = 3, we know the ratio of proportionality between these two triangles is 1/3. Thus, angle EFA must also be conrguent to angle EDH, and triangle EAF and EHD are similar. Note also that angle AEF is conrguent to angle HED. Since triangles EAB and GHD are equilateral (the side lengths are all congruent radii), then angle EAB is congruent to angle GHD. Then AB = AE = AG = GH = HD = 1, and EH = 3 Mark the point where these two circles intersect as point H. Construct circles with centers G and D with radius AB. Mark the point where this ray intersects circle A as point G. Claim:ĪF = 1/3 AB Note: For the proof of this construction you will also need to construct the ray EA. Mark the point where this segment intersects AB and call it F. Mark the upper point where circles A and B intersect as point E.
Construct ray BC and mark the point where it intersects circle C call it point D. Construct a circle with center C and radius AB. Mark the lower point where these two circles intersect as point C. To begin this construction, construct segment AB, a circle with center A and radius AB, and a circle with center B and radius AB. Trisection Construction #3: For a GSP script tool of this construction, click here. Since AD = 1 and AF = 3, we know that the ratio of proportionality is 1/3. Since line DG and FB are parallel, then we can know that triangle FAB is similar to triangle DAG. Your resulting figure should look similar to the one below. Mark the point where the line through D intersects segment AB and call it G. Construct line segment FB, then construct two lines parallel to this segment through points E and D. Mark the point where this circle intersects line CA and call it F. Construct a circle with center E and radius AD. Mark the point where this circle intersects line CA and call it E. Construct a circle with center D and radius AD. Mark another point on this line and call it D.
Then construct the line passing through point C and point A. To begin this construction, start with line segment AB and an arbitrary point C.
Note: For this construction, you need a line segment AB as well as arbitrary point C in order to get a trisected line segment. Trisection Construction #2: For a GSP script tool of this construction, click here. Since FG = 1, then we know AH must equal 1/3 and AH is thus equal to 1/3 of AB. Since AC = 1, CF = 3, and triangles CAH and CFG are similar, then we know the similarity ratio must be 1/3. Thus, angle CHA must also equal angle CGF, and triangles CAH and CFG are similar. Triangle CAB is congruent to triangle DFG (all of the side lengths are congruent radii), so angle CAB = angle DFG. Your resulting figure should look like the diagram below. Mark its intersection with line CE as point G. Construct a circle with center F and radius AB. Mark its intersection with ray CA as point F. Claim: AH = 1/3 AB Note: For the proof of this construction, you will also need to construct a circle with center D and radius AB. Mark the point where CE intersects AB as point H. Mark the point where DI intersects circle B as point E. Mark the point where this ray intersects circle A as point D.Įxtend AB into a ray and mark the point where this ray intersects circle B as point I. Construct ray CA, which is a diamter of circle A. Then construct a circle with center A and radius AB as well as a circle with center B and radius AB. To begin this construction, start with a segment AB. Trisecting a Line Segment by: Maggie Hendricks Trisection Construction #1: For a GSP script tool of this construction, click here.
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