construct a triangle ABC with AB=5.4cm BC=4.6cm

Steps, such as how to draw a perpendicular to a line, an angle bisector, or a perpendicular bisector of a segment, are skipped in here.   Mark point K.  From K, draw a horizontal ray rightwards about 12cm long, and mark the right end L.  Mark point B on KL about 4cm away from K.  Point A will be above BL.

From B, draw a vertical ray upwards about 8cm long, and mark the upper end V.   BV intersects KL at right angles, so ∠VBL=90°.  Bisect ∠VBL.   From B, draw the angle bisector upwards about 10cm long.  Mark  the upper end M above L, so ∠MBL=45°.   Note that the exact length of each line mentioned above is not important.

Mark point A on BM exactly 5.4cm away from B, and point C on BL exactly 4.6cm away from B.  Connect A to C.  △ABC is the required triangle with AB=5.4cm, BC=4.6cm and ∠ABC=45°.

Draw perpendicular bisectors of AB and BC, and mark the intersection of 2 bisectors O.  Draw a circle with its radius OA, and check if the circle passes thru points B and C.  CKD.  (Because △OAB,△OBC and △OCA are isoscelese, and share the sides of the same length: OA.)  Therefore, the circle O is the required one that circumscribes △ABC.　(AC=approx. 3.9cm, R=approx. 2.75cm)

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