Writing up constructions involves two steps: a construction or "recipe", where you state precisely all the steps of the construction, and secondly a proof that the construction does what you claim it does. Once you have done a construction once, you may refer to it everafter. So if you have proved that you have a constuction for an angle bisector, then after that you may use a step such as "Construct the bistector of angle PQX" and justify it by saying "Page 61, #2." CONSTRUCTIONS STEPS: The possiblities for the recipe steps are limited by the construction axioms. The first two of these types of recipe step are not listed in the drawing postulates. (Why? Maybe because they were considered too basic.) The remaining types of step are justified by the Postulates and by combining previously understood steps. (1) You may label (give a name to) any point you have drawn, or any point where two lines or circles intersect. (2) You can draw an arbitrary point (e.g., not on any existing line or circle). Be careful that you don't make any assumptions about such a point. For example, you can't just "draw" the midpoint of a line segment, though you may be able to do so with some sort of constructions. (3a) You may draw a line through two points (by Postulate 1). (3b) You may draw a line through a given point, but otherwise arbitrarily. Technically you can do this by combining steps of type 2 and 3a as follows. Call the given point P; choose an arbitrary point Q (type 2); draw the line PQ (type 3a). (3c) You may draw an arbitrary line. (Two steps of type 2 and one of type 3a). (4a) You may draw a circle with a given center and a given length as its radius (Postulate 3). Note: for a length to be "given", it must be the length between two given points on your paper. If someone has drawn a line segment you may not assume the endpoints are marked, and you may not draw in the endpoints. In fact if someone draws a segment on your paper with no points marked, the Greeks would consider that you have been given a line, not a line segment (since by Postulate 2, the segment may be extended in a line as far as you want). (4b) You may draw a circle centered at a given point with an arbitrary radius (Call the given point P; choose an arbitrary point Q and then draw the circle centered at P with radius PQ). (4c) You may draw an arbitrary circle (Choose P arbitrarily and proceed as in step 4b). PROOF STEPS: Some steps of a proof are simple assertions, justified as "by construction". For example, if you have drawn a circle through A with radius BC and then chosen a point D on the circle, the equality of lengths "AD = BC" is justified by construction. Later, when you use a complicated construction that you have previously proved, you may justify more complicated things by construction. For example, if you have in your construction "Step 5: Construct the bistector of angle PQX, call it QR" justified as "Page 61, #2", then a later step may "angle PQR = angle RQX", justified by construction (if there is any question as to which piece of the constuction justifies this, you should say, "by construction, step 5"). Justification steps involving theorems (from the Postulates and Theorems pages or one that you have proved yourself) should be handled similarly. For example, perhaps you may say "AB = CD", justified by "CPCTC". Here, you should make sure that a previous step has proved that two triangles are congruent, and that AB and CD are indeed corresponding sides of the two triangles. Thus if you are being careful, you'd say "CPCTC, step 7" where step 7 says something like "triangle ABX is congruent to triangle CDY". Self-check question: what will the justification for step 7 probably say? Answer: It will probably be one of the four congruence theorems such as SAS, and the equalities of the sides and angles you use ought to be stated as prior steps. In general, when you use a theorem, the hypotheses of the theorem should be established in previous steps. When you use one of the "common notions" you don't need to cite it. For example, if you are trying to establish the hypotheses of SSS so you can show that triangles ABC and ABE are congruent, you may write "AB = AB" without citing the fourth common notion "Things which coincide with one and other are equal". When you use a Basic Theorem it is probably best for now to cite it, even though they're pretty obvious.